Let \(\displaystyle x\) = 1st number

and 

\(\displaystyle x + 3\) = 2nd number

So the equation to solve becomes

 \(\displaystyle x(x + 3) = 108\)

or 

\(\displaystyle x^{2} + 3x - 108 = 0\)

We factor to solve the quadratic equation to get 9 and 12 and their sum is 21.

Let \(\displaystyle x\) = 1st odd number and \(\displaystyle x + 2\) = 2nd odd number.

So the equation to solve becomes

 \(\displaystyle x(x + 2) = 99\)

or

\(\displaystyle x^{2} + 2x - 99 = 0\)

Solving the quadratic equation by factoring gives 9 and 11, so the sum is 20

We can generate an equation for the number of home runs he has hit, x : x² - 50x = 50.  Reordering this, we get : x² - 50x - 50 = 0.  Using the quadratic equation: x = (-b± √(b²-4ac)) / (2a). In this case, a = 1, b = -50, c = -50.  Plugging in these values, we obtain the simplified equation, x = (50±51.96)/2.  Therefore, x = 50.98, -0.98.  Because it doesn't make sense to have a negative number of home runs, x = 50.98, which rounds up to 51 home runs. 

a = –3, b = –6.  The sum of a and b have to be equal to –9, and they have to multiply together to get +18. If a = –3 and b = –6, (–3) + (–6) = (–9) and (–3)(–6) = 18.  

Apply the quadratic formula directly to get [7 ± (49 – 4 * 11 * –8)^0.5]/22, → [7 ± (≈ 20)]/22

So our approximate answers are 27/22 and –13/22, and 27/22 is our answer.

Let \(\displaystyle x\) be defined as the lower number, and \(\displaystyle x + 3\) as the greater number.

We know that the first number times the second is \(\displaystyle 180\), so the equation to solve becomes \(\displaystyle x(x + 3) = 180\).

Distributing the \(\displaystyle x\) gives us a polynomial, which we can solve by factoring.

\(\displaystyle x^{2} + 3x - 180 = 0\)

\(\displaystyle -12*15=-180\) and \(\displaystyle -12+15=3\)

\(\displaystyle (x-12)(x+15)=0\)

\(\displaystyle x=12, -15\)

The question tells us that the integers are positive; therefore, \(\displaystyle x=12\).

If \(\displaystyle x=12\), and the second number is \(\displaystyle x + 3\), then the second number is \(\displaystyle 15\).

The sum of these numbers is \(\displaystyle 12+15=27\).

4y³ - 4y² = 8y

Divide by y and set equal to zero.

4y² - 4y – 8 = 0

(2y + 2)(2y – 4) = 0

2y + 2 = 0

2y = –2

y = –1

2y – 4 = 0

2y = 4

y = 2

You may use the quadratic formula (where \(\displaystyle a=9, b=-18, c=-27\)), which yields two answers, \(\displaystyle -1\) and \(\displaystyle 3\).

Since the only solution that appears in the answer list is \(\displaystyle 3\), we choose \(\displaystyle 3\).

Plug in y = 1. Then solve for x.

2x + y³ + xy² + y = x

2x + 1 + x + 1 = x

3x + 2 = x

2x = -2

x = -1

\(\displaystyle x\)-intercepts will occur when \(\displaystyle y=0\). This yields the equation

\(\displaystyle 0 = x^2 -4x +2\)

We need to use the quadratic formula where \(\displaystyle a=1\), \(\displaystyle b=-4\) and \(\displaystyle c=2\).

\(\displaystyle \frac{-b\pm(\sqrt{b^2-4ac})}{2a}\)

Plugging in our values:

\(\displaystyle \frac{4\pm\sqrt{16 - (4*1*2)}}{2}\)

Simplifying: 

\(\displaystyle \frac{4\pm \sqrt{8}}{2}\)

Simplifying:

\(\displaystyle \frac{4}{2} + \frac{\sqrt{8}}{2}\)

Simplifying:

\(\displaystyle 2\pm \frac{2\sqrt{2}}{2}\)

Finally:

 \(\displaystyle x = 2\pm \sqrt{2}\)