Multiple choice

Find the roots of the following equation: $ (x^2+x)(x^2+x-7)+10=0 $, then ( x 2 + x ) ( x 2 + x − 7 ) + 10 = 0 (x2+x)(x2+x−7)+10= , then

  1. $\displaystyle\,x\,=\,\frac{-1\,\pm\,\sqrt{21}}{2},\,1,\,-2$
  2. $\displaystyle\,x\,=\,\frac{-1\,\pm\,\sqrt{23}}{2},\,1,\,-2$
  3. $\displaystyle\,x\,=\,\frac{-1\,\pm\,\sqrt{21}}{2},\,-1,\,-2$
  4. $\displaystyle\,x\,=\,\frac{-1\,\pm\,\sqrt{21}}{2},\,1,\,2$
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A Correct answer
Explanation

Let y = x^2 + x. Then y(y - 7) + 10 = 0, so y = 5 or y = 2. These give x = (-1 +/- sqrt(21))/2 and x = 1, -2, respectively.

AI explanation

Let m = x^2 + x, which rewrites the equation as m(m - 7) + 10 = 0 or m^2 - 7m + 10 = 0. Factoring this gives (m - 5)(m - 2) = 0, meaning m equals 5 or 2. Solving x^2 + x = 5 using the quadratic formula gives x = (-1 plus or minus the square root of 21)/2, and solving x^2 + x = 2 yields (x + 2)(x - 1) = 0, giving x = 1 or -2.

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