Answer:
Step-by-step explanation:
(1) [tex]f(x) = (x-1)^{2} - 36[/tex]
First, let's expand the expression:
[tex]f(x) = x^{2} -2x + 1 - 36[/tex]
[tex]f(x) = x^{2} - 2x - 35[/tex]
Next, let's factor the quadratic:
[tex]f(x) = (x - 7)(x + 5)[/tex]
Finally, set the equation equal to [tex]0[/tex] to find the solutions
[tex]0 = (x - 7)(x + 5)[/tex]
[tex]x = -5, 7[/tex]
(2) During the step [tex](x - 5)^{2} = 36[/tex], when you take the square root of both sides, you have two equations, rather than one:
[tex]x - 5 = 6[/tex] and [tex]x - 5 = -6[/tex], so the solutions should be [tex]x = -1, 11[/tex]
(3)
[tex]\frac{1}{4}(x + 5)^{2} - 1 = 3[/tex]
Add [tex]1[/tex] to both sides
[tex]\frac{1}{4}(x + 5)^{2} = 4[/tex]
Multiply both sides by [tex]4[/tex]
[tex](x + 5)^{2} = 16[/tex]
Take the square root of both sides
[tex]x + 5 = [/tex]±[tex]4[/tex]
Subtract [tex]5[/tex] from both sides
[tex]x = -9, -1[/tex]
