Quadratic Equations and Inequalities - Algebra II

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

Finish out the solution:

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

Finish out the solution:

Make sure you keep track of negative signs when doing FOIL, especially when doing the Outer and Inner steps.

When the 2 terms differ only in their sign, the -term drops out from the final product.

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

Finish out the solution:

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Take the square root of both sides:

Finish out the solution:

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Use a calculator to determine the final values.

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Use a calculator to determine the final values.

To solve by completing the square, you should first take the numerical coefficient to the “right side” of the equation:

Then, divide the middle coefficient by 2:

Square that and add it to both sides:

Now, you can easily factor the quadratic:

Your next step would be to take the square root of both sides. At this point, however, you know that you cannot solve the problem. When you take the square root of both sides, you will be forced to take the square root of . This is impossible (at least in terms of real numbers), meaning that this problem must have no real solution.

In order to evaluate one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.

Multiply terms by way of FOIL method.

Now multiply and simplify.

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Separate this expression into two fractions and simplify to determine the final values.

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Separate this expression into two fractions and simplify to determine the final values.

Use the FOIL method (First, Outer, Inner, Last):

Put all of these terms together:

Combine like terms:

The discriminant is found using the formula . In this case:

To find the discriminant, use the formula . In this case:

To determine the amount of roots a given quadratic function has, we must find the discriminant, which for

is equal to

If d is negative, then we have two roots that are complex conjugates of one another. If d is positive, than we have two real roots, and if d is equal to zero, then we have only one real root.

Using our function and the formula above, we get

Thus, the function has only one real root.

Identify the coefficients for the polynomial .

Write the expression for the discriminant. This is the expression inside the square root from the quadratic formula.

Substitute the numbers.

The answer is:

In general, the discriminant is .

In this particual case .

Plug in these three values and simplify:

Given the quadratic equation of

The formula for the discriminant is (remember this as a part of the quadratic formula?)

Plugging in values to the discriminant equation:

So the discriminant is 57. What does that mean for our solutions? Since it is a positive number, we know that we will have 2 real solutions. So the answer is:

57, 2 real solutions