To find the zeroes of the quadratic equation [tex]\( y = -16x^2 + 64x + 80 \)[/tex], follow these steps:
1. Identify the coefficients:
The given quadratic equation is [tex]\( y = -16x^2 + 64x + 80 \)[/tex].
Here, the coefficients are:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 64 \)[/tex]
- [tex]\( c = 80 \)[/tex]
2. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 64^2 - 4(-16)(80)
\][/tex]
[tex]\[
\Delta = 4096 + 5120
\][/tex]
[tex]\[
\Delta = 9216
\][/tex]
3. Apply the quadratic formula:
The quadratic formula to find the roots of [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Using the discriminant [tex]\(\Delta = 9216\)[/tex], solve for the roots:
[tex]\[
x = \frac{-64 \pm \sqrt{9216}}{2(-16)}
\][/tex]
4. Calculate the square root of the discriminant:
[tex]\[
\sqrt{9216} = 96
\][/tex]
5. Find the two solutions:
[tex]\[
x_1 = \frac{-64 + 96}{-32} = \frac{32}{-32} = -1
\][/tex]
[tex]\[
x_2 = \frac{-64 - 96}{-32} = \frac{-160}{-32} = 5
\][/tex]
6. State the zeroes:
The zeroes of the quadratic equation [tex]\( y = -16x^2 + 64x + 80 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
These steps lead to the following results:
- Discriminant: [tex]\( 9216 \)[/tex]
- First zero: [tex]\( x = -1 \)[/tex]
- Second zero: [tex]\( x = 5 \)[/tex]
So, the zeroes of the quadratic equation [tex]\( y = -16x^2 + 64x + 80 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
