Multiple choice

Which of the following is a solution of the quadratic equation: $x^{2}-b^{2}=a\left ( 2x-a \right )$?

  1. $a + b$
  2. $2b - a$
  3. $ab$
  4. $\dfrac{a}{b}$
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A Correct answer
Explanation

x^2 - b^2 = 2ax - a^2 => x^2 - 2ax + a^2 - b^2 = 0 => (x-a)^2 - b^2 = 0 => (x-a-b)(x-a+b) = 0. Roots are a+b and a-b.

AI explanation

Expand the right side of the equation to get x^2 - b^2 = 2ax - a^2, then rearrange all terms to form the standard quadratic equation x^2 - 2ax + (a^2 - b^2) = 0. Factoring this as the difference of squares gives (x - a)^2 - b^2 = 0, which becomes (x - a + b)(x - a - b) = 0. The roots of the equation are (a - b) and (a + b). Therefore, the solution is a + b.

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