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:
- Start with the equation: \( y = x^2 + 2x - 8 \).
- 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:
- \( x = \frac{4}{2} = 2 \)
- \( x = \frac{-8}{2} = -4 \)
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:
- Start with \( y = x^2 + 2x - 8 \).
- Take the coefficient of \( x \), which is \( 2 \). Half of it is \( 1 \), and squaring it gives \( 1 \).
- 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: \( y = (x + 1)^2 - 9 \)