Question

What equation matches y = x^2 + 2x - 8

Answer 1

The equation \( y = x^2 + 2x - 8 \) is a quadratic equation. It can be rewritten in various forms, such as the vertex form or factored form.

To find the factored form, we can factor the quadratic:

1. Start with the equation: \( y = x^2 + 2x - 8 \).
2. We want to express it in the form \( y = (x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots of the equation.

To find the roots, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 2 \), and \( c = -8 \).

Calculating the discriminant:
\[
b^2 - 4ac = 2^2 - 4(1)(-8) = 4 + 32 = 36
\]

Now applying the quadratic formula:
\[
x = \frac{-2 \pm \sqrt{36}}{2(1)} = \frac{-2 \pm 6}{2}
\]

Calculating the two roots:
1. \( x = \frac{4}{2} = 2 \)
2. \( x = \frac{-8}{2} = -4 \)

Thus, the roots are \( x = 2 \) and \( x = -4 \).

Now we can write the factored form:
\[
y = (x - 2)(x + 4)
\]

Alternatively, we can also express the quadratic in vertex form, which is useful for graphing. The vertex form of a quadratic is given by:
\[
y = a(x - h)^2 + k
\]
where \((h,k)\) is the vertex.

To convert \( y = x^2 + 2x - 8 \) to vertex form, we can complete the square:

1. Start with \( y = x^2 + 2x - 8 \).
2. Take the coefficient of \( x \), which is \( 2 \). Half of it is \( 1 \), and squaring it gives \( 1 \).
3. Rewrite the equation:
\[
y = (x^2 + 2x + 1) - 1 - 8
\]
\[
y = (x + 1)^2 - 9
\]

Now we have the vertex form:
\[
y = (x + 1)^2 - 9
\]
The vertex of the parabola is at the point \( (-1, -9) \).

In summary, the equation \( y = x^2 + 2x - 8 \) can be expressed in:
- Factored form: \( y = (x - 2)(x + 4) \)
- Vertex form: \( y = (x + 1)^2 - 9 \)