Question

what are the solutions of the quadratic equation ((x - 8)^2 - 13(x - 8) + 30 = 0)? use (u) substitution to solve.
(x = -11) and (x = -18)
(x = -2) and (x = 5)
(x = 2) and (x = -5)
(x = 11) and (x = 18)

Explanation:

Step1: Define substitution

Let \( u = x - 8 \). Then the equation becomes \( u^2 - 13u + 30 = 0 \).

Step2: Factor the quadratic

Factor \( u^2 - 13u + 30 \). We need two numbers that multiply to 30 and add to -13. Those numbers are -10 and -3. So, \( (u - 10)(u - 3) = 0 \).

Step3: Solve for u

Set each factor equal to zero: \( u - 10 = 0 \) or \( u - 3 = 0 \), so \( u = 10 \) or \( u = 3 \).

Step4: Substitute back for x

Since \( u = x - 8 \), for \( u = 10 \): \( x - 8 = 10 \), so \( x = 18 \). For \( u = 3 \): \( x - 8 = 3 \), so \( x = 11 \).

Answer:

x = 11 and x = 18