Given that

Question

(𝑥 − 𝐴)2 + 𝐵 = 𝑥2 − 10𝑥 +
29, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐴 + 𝐵
A. -9
B.-1
C.8
D. 9

Answer 1

Expanding the left-hand side of the given equation, we have:

(𝑥 − 𝐴)2 + 𝐵 = 𝑥2 − 2𝑥𝐴 + 𝐴2 + 𝐵

Setting this equal to 𝑥2 − 10𝑥 + 29, we get:

𝑥2 − 2𝑥𝐴 + 𝐴2 + 𝐵 = 𝑥2 − 10𝑥 + 29

Simplifying and rearranging, we get:

2𝑥𝐴 − 𝐴2 − 10𝑥 + 𝐵 − 29 = 0

This is a quadratic equation in 𝑥, with coefficients that depend on 𝐴 and 𝐵. We can use the quadratic formula to find the solutions for 𝑥:

𝑥 = [10 ± sqrt(100 - 4(2𝐴 - 𝐵 + 29 - 𝐴2))]/(4)

Simplifying under the square root, we get:

100 - 8𝐴 + 2𝐵 - 4𝐴2 - 116 = -4(𝐴 - 4)2

Substituting this back into the formula for 𝑥, we get:

𝑥 = [10 ± 2sqrt(-4(𝐴 - 4)2)]/4 = 5 ± i(𝐴 - 4)/2

The real part of these solutions is 𝑥 = 5, so we need to choose the value of 𝐴 and 𝐵 that make the imaginary part vanish. This happens when:

𝐴 - 4 = 0, so 𝐴 = 4

-𝐴2 + 𝐵 - 29 = 0, so 𝐵 = 𝐴2 - 29 = -13

Therefore, the value of 𝑓(𝐴 + 𝐵) is:

𝑓(4 + (-13)) = 𝑓(-9)

Substituting into the original equation, we get:

(𝑥 − 4)2 - 13 = 𝑥2 − 10𝑥 + 29

Simplifying and rearranging, we get:

𝑥2 - 12𝑥 + 56 = 0

Factoring, we get:

(𝑥 - 4)(𝑥 - 8) = 0

Therefore, the solutions for 𝑥 are 𝑥 = 4 and 𝑥 = 8. The value we are looking for is the negative of the sum of these solutions:

- (4 + 8) = -12

Therefore, the answer is (E) -12.