Sure, let's solve the quadratic function [tex]\( y = 64 - 16x^2 \)[/tex] step-by-step using the square root method.
1. Set the function equal to zero to find the x-intercepts:
We start by setting [tex]\( y \)[/tex] to zero because we want to find the values of [tex]\( x \)[/tex] where the function intersects the x-axis.
[tex]\[
0 = 64 - 16x^2
\][/tex]
2. Isolate the term with [tex]\( x^2 \)[/tex]:
Move 64 to the left side of the equation to get:
[tex]\[
16x^2 = 64
\][/tex]
3. Divide both sides by 16 to solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{64}{16}
\][/tex]
4. Simplify the right side of the equation:
[tex]\[
x^2 = 4
\][/tex]
5. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
When we take the square root of both sides, we must consider both the positive and negative roots.
[tex]\[
x = \pm \sqrt{4}
\][/tex]
6. Simplify the square roots:
[tex]\[
x = \pm 2
\][/tex]
Therefore, the solutions to the quadratic function [tex]\( y = 64 - 16x^2 \)[/tex] are:
[tex]\[
x = 2 \quad \text{and} \quad x = -2
\][/tex]
So, the final result is:
[tex]\[
x = \pm 2
\][/tex]
