Question

question 5 (mandatory) (1 point) saved
determine the zeros for the function ( f(x) = -4(x - 6)^2 + 6 ) by isolating ( x ).

a) ( (0, 7.22) ) and ( (0, 4.78) )

b) ( (4.78, 0) ) and ( (7.22, 0) )

c) ( (24, 0) ) and ( (6, 0) )

d) ( (6, 0) ) and ( (-6, 0) )

Explanation:

Step1: Set \( f(x) = 0 \)

We start by setting the function \( f(x)=-4(x - 6)^{2}+6 \) equal to zero, so we have the equation:

$$ 0=-4(x - 6)^{2}+6 $$

Step2: Isolate the squared term

First, we add \( 4(x - 6)^{2} \) to both sides of the equation:

$$ 4(x - 6)^{2}=6 $$

Then, we divide both sides by 4:

$$ (x - 6)^{2}=\frac{6}{4}=\frac{3}{2} $$

Step3: Take square roots

Take the square root of both sides. Remember that when we take the square root, we get two solutions (positive and negative square roots):

$$ x - 6=\pm\sqrt{\frac{3}{2}} $$

Simplify \( \sqrt{\frac{3}{2}}\approx\sqrt{1.5}\approx1.22 \)

Step4: Solve for \( x \)

For the positive square root:

$$ x=6 + \sqrt{\frac{3}{2}}\approx6 + 1.22 = 7.22 $$

For the negative square root:

$$ x=6-\sqrt{\frac{3}{2}}\approx6 - 1.22 = 4.78 $$

The zeros of the function are the \( x \)-intercepts, which have the form \( (x,0) \). So the zeros are \( (4.78,0) \) and \( (7.22,0) \)

Answer:

b) (4.78, 0) and (7.22, 0)