Multiple choice

Number of rational roots of the equation $\sqrt{5x^{2}-6x+8}-\sqrt{5x^{2}-6x-7}=1$ is/are

  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let y = 5x^2 - 6x. The equation becomes sqrt(y+8) - sqrt(y-7) = 1. Squaring both sides gives (y+8) + (y-7) - 2*sqrt((y+8)(y-7)) = 1, which simplifies to 2y + 1 - 1 = 2*sqrt(y^2 + y - 56), or y = sqrt(y^2 + y - 56). Squaring again, y^2 = y^2 + y - 56, so y = 56. Substituting back, 5x^2 - 6x - 56 = 0. Solving via quadratic formula, x = (6 +/- sqrt(36 - 4*5*(-56)))/10 = (6 +/- sqrt(1156))/10 = (6 +/- 34)/10. The roots are 4 and -2.8. Both are rational.

AI explanation

Squaring both sides of the equation gives (5x^2 - 6x + 8) + (5x^2 - 6x - 7) - 2 * sqrt((5x^2 - 6x + 8)(5x^2 - 6x - 7)) = 1. Simplifying this results in 10x^2 - 12x - 2 * sqrt((5x^2 - 6x)^2 + (5x^2 - 6x) - 56) = 0. Let t = 5x^2 - 6x, giving 2t = 2 * sqrt(t^2 + t - 56), which further simplifies to 3t^2 - 3t - 224 = 0. Solving for t yields t = 28/3 or t = -8, and substituting t = -8 gives 5x^2 - 6x + 8 = 0 which has two distinct rational roots, while t = 28/3 has irrational roots. There are exactly 2 rational roots.

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