Question

Solve the system of equations.

y=2x2−9


y=x2−10x−25


Enter the solution with the least x-value first.

(1 point)
The solutions are
,
, and
,
.

Answer 1

To solve the system of equations given by:

1. \( y = 2x^2 - 9 \)
2. \( y = x^2 - 10x - 25 \)

we can set the two equations equal to each other:

\[
2x^2 - 9 = x^2 - 10x - 25
\]

Now, let's rearrange this equation:

\[
2x^2 - x^2 + 10x - 9 + 25 = 0
\]

This simplifies to:

\[
x^2 + 10x + 16 = 0
\]

Next, we can factor this quadratic equation. We need two numbers that multiply to 16 and add to 10. Those numbers are 2 and 8. Thus, we can factor the equation:

\[
(x + 2)(x + 8) = 0
\]

Setting each factor to zero gives us:

\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]

Now that we have the \( x \)-values, we can substitute them back into either original equation to find the corresponding \( y \)-values. Let's use the first equation \( y = 2x^2 - 9 \).

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

\[
y = 2(-2)^2 - 9 = 2(4) - 9 = 8 - 9 = -1
\]

So, one solution is \((-2, -1)\).

2. For \( x = -8 \):

\[
y = 2(-8)^2 - 9 = 2(64) - 9 = 128 - 9 = 119
\]

So, the other solution is \((-8, 119)\).

Now, we can list the solutions in the specified format:

The solutions are \((-8, 119)\) and \((-2, -1)\).

Hence, according to the instruction to enter the solution with the least \( x \)-value first, the final answer is:

The solutions are \(-8\), \(119\), and \(-2\), \(-1\).