Question

Solve the system of equations.
y=x2+32x+44
y=15x+28
Write the coordinates in exact form. Simplify all fractions and radicals.
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Answer 1

To solve the system of equations

1. \( y = x^2 + 32x + 44 \)
2. \( y = 15x + 28 \)

we will set the two equations equal to each other:

\[
x^2 + 32x + 44 = 15x + 28
\]

Next, we will rearrange the equation by moving all terms to one side:

\[
x^2 + 32x + 44 - 15x - 28 = 0
\]

Simplifying this gives:

\[
x^2 + 17x + 16 = 0
\]

Now, we will factor the quadratic equation:

\[
(x + 1)(x + 16) = 0
\]

Setting each factor equal to zero gives us the solutions for \( x \):

1. \( x + 1 = 0 \) → \( x = -1 \)
2. \( x + 16 = 0 \) → \( x = -16 \)

Next, we substitute each \( x \) value back into one of the original equations to find the corresponding \( y \) values. We can use the linear equation \( y = 15x + 28 \).

1. For \( x = -1 \):

\[
y = 15(-1) + 28 = -15 + 28 = 13
\]

Thus, one solution is \( (-1, 13) \).

2. For \( x = -16 \):

\[
y = 15(-16) + 28 = -240 + 28 = -212
\]

Thus, the other solution is \( (-16, -212) \).

Now we have the solutions for the system of equations. The coordinates in exact form are:

\[
(-1, 13) \quad \text{and} \quad (-16, -212)
\]