Seeing Structure in Expressions - Algebra

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Question

Identify the following series as arithmetic, geometric, or neither.

100, 200, 300, 400, 500...

Answer

This is an arithmetic sequence. Note that the same number (100) is added to each value in the set to give us our next number:

100

100 + 100 = 200

200 + 100 = 300

300 + 100 = 400

400 + 100 = 500

When the same value is added to each term to determine the next one, that is an arithmetic sequence.

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Question

1, 9, 81, 729, 6561...

Is the sequence above arithmetic, geometric, or neither?

Answer

The series is geometric, which means that to get from one value to the next you always multiply by the same value. Here to get from one value to the next, you multiply by 9 each time:

1 x 9 = 9

9 x 9 = 81

81 x 9 = 729

729 x 9 = 6561

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Question

What type of sequence describes the following set of numbers?

3, 15, 75, 375, 1875

Answer

This is a geometric series. The same value, 5, is multiplied by each value to get to the next:

3 x 5 = 15

15 x 5 = 75

75 x 5 = 375

375 x 5 = 1875

When each term in a series is equal to the previous term multiplied by the same factor - in this case 5 - that's a geometric series.

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+4x+7=0\Rightarrow (x+2)^2+2^2+7=0+2^2$

When simplified the new function is,

$\(x+2)^2+4+7=4 \(x+2)^2+11=4 \(x+2)^2=-7$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=(-2)^2+4(-2)+7 \y=4-8+7 \y=3$

Therefore the minimum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+2x-4=0\Rightarrow (x+1)^2+1^2-4=0+1^2$

When simplified the new function is,

$\(x+1)^2-4+1 \(x+1)^2-3$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+1=0 \x+1-1=0-1 \x=-1$

From here, substitute the the value into the original function.

$\y=(-1)^2+2(-1)-4 \y=1-2-4 \y=-5$

Therefore the minimum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$x^2+8x-3\Rightarrow (x+4)^2-3+4^2$

When simplified the new function is,

$\(x+4)^2-3+16\(x+4)^2+13$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+4=0 \x+4-4=0-4 \x=-4$

From here, substitute the the value into the original function.

$\y=(-4)^2+8(-4)-3 \y=16-32-3 \y=-19$

Therefore the minimum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$x^2+6x+2\Rightarrow (x+3)^2+6+3^2$

When simplified the new function is,

$\(x+3)^2+2+9\(x+3)^2+11$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=(-3)^2+6(-3)+2 \y=9-18+2 \y=-7$

Therefore the minimum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -4$

Now divide the middle term coefficient by two.

$\frac{-4}{2}=-2$

From here write the function with the perfect square.

$x^2-4x+1\Rightarrow (x-2)^2+1+(-2)^2$

When simplified the new function is,

$\(x-2)^2+1+4\(x-2)^2+5$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x-2=0 \x-2+2=0+2\x=2$

From here, substitute the the value into the original function.

$\y=(2)^2-4(2)+1 \y=4-8+1 \y=-3$

Therefore the minimum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -6$

Now divide the middle term coefficient by two.

$\frac{-6}{2}=-3$

From here write the function with the perfect square.

$x^2-6x-6\Rightarrow (x-3)^2-6+(-3)^2$

When simplified the new function is,

$\(x-3)^2-6+9\(x-3)^2+3$

Since the term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x-3=0 \x-3+3=0+3\x=3$

From here, substitute the the value into the original function.

$\y=(3)^2-6(3)-6 \y=9-18-6 \y=-15$

Therefore the minimum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square.

$-(x^2+2x+1)\Rightarrow-((x+1)^2+1+1)$

When simplified the new function is,

$\-((x+1)^2+2)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+1=0 \x+1-1=0-1 \x=-1$

From here, substitute the the value into the original function.

$\y=-(-1)^2-2(-1)-1 \y=-1+2-1 \y=0$

Therefore the maximum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+1)\Rightarrow-((x+2)^2+1+2^2)$

When simplified the new function is,

$\-((x+2)^2+5)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=-(-2)^2-4(-2)-1 \y=-4+8-1 \y=3$

Therefore the maximum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$-(x^2+6x+1)\Rightarrow-((x+3)^2+1+3^2)$

When simplified the new function is,

$\-((x+3)^2+10)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=-(-3)^2-6(-3)-1 \y=-9+18-1 \y=8$

Therefore the maximum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$-(x^2+8x+2)\Rightarrow-((x+4)^2+2+4^2)$

When simplified the new function is,

$\-((x+4)^2+18)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+4=0 \x+4-4=0-4 \x=-4$

From here, substitute the the value into the original function.

$\y=-(-4)^2-8(-4)-2 \y=-16+32-2 \y=14$

Therefore the maximum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+5)\Rightarrow-((x+2)^2+5+2^2)$

When simplified the new function is,

$\-((x+2)^2+9)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+2=0 \x+2-2=0-2 \x=-2$

From here, substitute the the value into the original function.

$\y=-(-2)^2-4(-2)-5 \y=-4+8-5 \y=-1$

Therefore the maximum value occurs at the point .

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Question

Complete the square to calculate the maximum or minimum point of the given function.

Answer

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

First factor out a negative one.

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$-(x^2+6x+5)\Rightarrow-((x+3)^2+5+3^2)$

When simplified the new function is,

$\-((x+3)^2+14)$

Since the term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\x+3=0 \x+3-3=0-3 \x=-3$

From here, substitute the the value into the original function.

$\y=-(-3)^2-6(-3)-5 \y=-9+18-5 \y=4$

Therefore the maximum value occurs at the point .

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Question

Identify the following sequence as arithmetic, geometric, or neither.

Answer

To identify the sequence as either arithmetic or geometric, first recall the difference between the two. An arithmetic sequence means that there is a common difference between the terms. If a sequence is geometric, then there is a common ratio between the terms.

Looking at the given sequence,

subtract the first term from the second term to find the common difference.

$\text{Common Difference: }-1-(-3)=-1+3=2$

From here, add the common difference to the second term. If the sum is the third term, then the sequence is arithmetic.

Adding the common difference to each term in the sequence results in the next term in the sequence which makes this particular sequence, arithmetic.

$\-1+2=1 \1+2=3 \3+2=5$

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Question

Identify the following sequence as arithmetic, geometric, or neither.

Answer

To identify the sequence as either arithmetic or geometric, first recall the difference between the two. An arithmetic sequence means there exists a common difference between the terms. If a sequence is geometric, then there exists a common ratio between the terms.

Looking at the given sequence,

subtract the first term from the second term to find the common difference.

$\text{Common Difference: }-4-(-10)=-4+10=6$

From here, add the common difference to the second term. If the sum is the third term, then the sequence is arithmetic.

Adding the common difference to each term in the sequence results in the next term in the sequence which makes this particular sequence, arithmetic.

$\-4+6=2 \2+6=8 \8+6=14$

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Question

Identify the following sequence as arithmetic, geometric, or neither.

Answer

To identify the sequence as either arithmetic or geometric, first recall the difference between the two. An arithmetic sequence means there exists a common difference between the terms. If a sequence is geometric, then there exists a common ratio between the terms.

Looking at the given sequence,

subtract the first term from the second term to find the common difference.

$\text{Common Difference: }12-2=10$

From here, add the common difference to the second term. If the sum is the third term, then the sequence is arithmetic.

Adding the common difference to each term in the sequence results in the next term in the sequence which makes this particular sequence, arithmetic.

$\12+10=22 \22+10=32 \32+10=42$

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Question

Identify the following sequence as arithmetic, geometric, or neither.

Answer

To identify the sequence as either arithmetic or geometric, first recall the difference between the two. An arithmetic sequence means there exists a common difference between the terms. If a sequence is geometric, then there exists a common ratio between the terms.

Looking at the given sequence,

subtract the first term from the second term to find the common difference.

$\text{Common Difference: }-4-(-10)=-4+10=6$

From here, add the common difference to the second term.

If adding the common difference to each term in the sequence results in next term then the sequence is arithmetic.

All terms except for the fourth term follow this therefore, the sequence is neither arithmetic nor geometric.

$\-4+6=2 \2+6=8 \7+6=13$

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Question

Identify the following sequence as arithmetic, geometric, or neither.

Answer

To identify the sequence as either arithmetic or geometric, first recall the difference between the two. An arithmetic sequence means there exists a common difference between the terms. If a sequence is geometric, then there exists a common ratio between the terms.

Looking at the given sequence,

subtract the first term from the second term to find the common difference.

$\text{Common Difference: }12-2=10$

From here, add the common difference to the second term. If the sum is the third term, then the sequence is arithmetic.

If adding the common difference to each term in the sequence results in next term then the sequence is arithmetic.

All terms except for the third term follow this therefore, the sequence is neither arithmetic nor geometric.

$\12+10=22 \24+10=34 \32+10=42$

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