Given that
(๐ฅ โ ๐ด)2 + ๐ต = ๐ฅ2 โ 10๐ฅ +
29, ๐๐๐๐ ๐กโ๐ ๐ฃ๐๐๐ข๐ ๐๐ ๐ด + ๐ต
A. -9
B.-1
C.8
D. 9
Expanding the left-hand side of the given equation, we have:
(๐ฅ โ ๐ด)2 + ๐ต = ๐ฅ2 โ 2๐ฅ๐ด + ๐ด2 + ๐ต
Setting this equal to ๐ฅ2 โ 10๐ฅ + 29, we get:
๐ฅ2 โ 2๐ฅ๐ด + ๐ด2 + ๐ต = ๐ฅ2 โ 10๐ฅ + 29
Simplifying and rearranging, we get:
2๐ฅ๐ด โ ๐ด2 โ 10๐ฅ + ๐ต โ 29 = 0
This is a quadratic equation in ๐ฅ, with coefficients that depend on ๐ด and ๐ต. We can use the quadratic formula to find the solutions for ๐ฅ:
๐ฅ = [10 ยฑ sqrt(100 - 4(2๐ด - ๐ต + 29 - ๐ด2))]/(4)
Simplifying under the square root, we get:
100 - 8๐ด + 2๐ต - 4๐ด2 - 116 = -4(๐ด - 4)2
Substituting this back into the formula for ๐ฅ, we get:
๐ฅ = [10 ยฑ 2sqrt(-4(๐ด - 4)2)]/4 = 5 ยฑ i(๐ด - 4)/2
The real part of these solutions is ๐ฅ = 5, so we need to choose the value of ๐ด and ๐ต that make the imaginary part vanish. This happens when:
๐ด - 4 = 0, so ๐ด = 4
-๐ด2 + ๐ต - 29 = 0, so ๐ต = ๐ด2 - 29 = -13
Therefore, the value of ๐(๐ด + ๐ต) is:
๐(4 + (-13)) = ๐(-9)
Substituting into the original equation, we get:
(๐ฅ โ 4)2 - 13 = ๐ฅ2 โ 10๐ฅ + 29
Simplifying and rearranging, we get:
๐ฅ2 - 12๐ฅ + 56 = 0
Factoring, we get:
(๐ฅ - 4)(๐ฅ - 8) = 0
Therefore, the solutions for ๐ฅ are ๐ฅ = 4 and ๐ฅ = 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.
To find the value of A + B, we need to compare the expanded form of the given equation with the standard form of a quadratic equation, which is ax^2 + bx + c.
Given equation: (x - A)^2 + B = x^2 - 10x + 29
Comparing all the terms:
(x - A)^2 + B = x^2 - 10x + 29
Expanding (x - A)^2:
x^2 - 2Ax + A^2 + B = x^2 - 10x + 29
Comparing like terms:
-2Ax + A^2 + B = -10x + 29
Comparing the x term on both sides, we get:
-2A = -10
Solving for A:
A = -10 / -2 = 5
Now, substituting the value of A back into the equation, we get:
-10x + A^2 + B = -10x + 29
A^2 + B = 29
Substituting the value of A (5) into the equation:
5^2 + B = 29
25 + B = 29
Subtracting 25 from both sides:
B = 4
Now, calculating A + B:
A + B = 5 + 4 = 9
Therefore, the value of A + B is 9. So, the correct answer is D. 9.