Multiple choice

If a and b are the roots of the equation $\displaystyle { x }^{ 2 }-5x+6=0$, find the value of $\displaystyle \left( { a }^{ 2 }-{ b }^{ 2 } \right) $.

  1. $\displaystyle \pm 3$
  2. $\displaystyle \pm 5$
  3. $\displaystyle \pm 4$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots of x^2 - 5x + 6 = 0 are 2 and 3. So a=2, b=3 or vice-versa. a^2 - b^2 = 4 - 9 = -5 or 9 - 4 = 5. Thus, the value is +/- 5.

AI explanation

Using Vieta's formulas for x^2 - 5x + 6 = 0, the sum of the roots a + b equals 5 and the product ab equals 6. The difference of squares factors as a^2 - b^2 = (a - b)(a + b), so we must first find the value of (a - b). We know that (a - b)^2 = (a + b)^2 - 4ab = 5^2 - 4(6) = 25 - 24 = 1, which means a - b = 1 or a - b = -1. Therefore, a^2 - b^2 equals 5 * 1 or 5 * (-1), resulting in a value of 5 or -5.

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