Question

If \(x^2+2 x y+y^2=64\) and \(y-x=12\), which of the following could be the value of \(x\) ? A) \(-10\) B) \(-4\) C) 2 D) 10

Short answer

The correct answer is: \(x = -10\) (Option A).

Step-by-step solution

Make y the subject in the second equation

In the equation \(y - x = 12\), add x to both sides to get: \(y = x + 12\)

Substitute the expression for y in the first equation

Replace \(y\) in the equation \(x^2 + 2xy + y^2 = 64\) with the expression from Step 1: \(x^2 + 2x(x + 12) + (x + 12)^2 = 64\)

Simplify the equation

Expand the equation and combine like terms: \(x^2 + 2x^2 + 24x + x^2 + 24x + 144 = 64\) Combine the x terms: \(4x^2 + 48x + 144 = 64\)

Further simplify the equation

To make the equation easier to solve, we can subtract 64 from both sides: \(4x^2 + 48x + 80 = 0\) Now, divide the entire equation by 4: \(x^2 + 12x + 20 = 0\)

Factor the quadratic equation

To solve for x, we need to factor the quadratic equation: \((x + 2)(x + 10) = 0\)

Find the values of x

Set each factor equal to 0 and solve for x: \(x + 2 = 0 \Rightarrow x = -2\) \(x + 10 = 0 \Rightarrow x = -10\)

Check the possible value of x

We have two potential solutions for x: -2 and -10. Let's see which of these is among the given options: A) \(x = -10\) - This value is among our solutions. B) \(x = -4\) - This value is not among our solutions. C) \(x = 2\) - This value is not among our solutions. D) \(x = 10\) - This value is not among our solutions. So, the correct answer is: \(x = -10\) (Option A).