Which of the following equations and solutions is matched correctly?(1 point) Responses −x2−64=0 , x=−8, 8 negative x squared minus 64 equals 0 , x equals negative 8 comma 8 −x2−x−70=0 , x=−10, 7 negative x squared minus x minus 70 is equal to 0 , x equals negative 10 comma 7 4x2−3x2+10x=−25 , x=−5 4 x squared minus 3 x squared plus 10 x equals negative 25 , x equals negative 5 x2+6x=112 , x=−8, 14

Answer 1

Let's analyze each pair of equations and their solutions to determine which is correctly matched.

1. Equation: \(-x^2 - 64 = 0\)

   Solution: \(x = -8, 8\)

   Rearranging the equation: \(-x^2 = 64 \implies x^2 = -64\) which has no real solutions. Therefore, this pair is mismatched.

2. Equation: \(-x^2 - x - 70 = 0\)

   Solution: \(x = -10, 7\)

   Rearranging the equation: \(-x^2 - x - 70 = 0 \implies x^2 + x + 70 = 0\) which has a negative discriminant \(b^2 - 4ac = 1 - 280 = -279\), resulting in no real solutions. Therefore, this pair is also mismatched.

3. Equation: \(4x^2 - 3x^2 + 10x = -25\)

   Solution: \(x = -5\)

   Rearranging the equation: \(4x^2 - 3x^2 + 10x + 25 = 0 \implies x^2 + 10x + 25 = 0 \implies (x + 5)^2 = 0\) gives \(x = -5\) (double root). This pair is correctly matched.

4. Equation: \(x^2 + 6x = 112\)

   Solution: \(x = -8, 14\)

   Rearranging the equation: \(x^2 + 6x - 112 = 0\). The roots can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-6 \pm \sqrt{36 + 448}}{2} = \frac{-6 \pm 22}{2}\), resulting in the solutions \(x = 8\) and \(x = -14\). Thus, this pair is mismatched.

Based on the analysis, only the third pair is matched correctly:

Correctly matched equation and solution:
\(4x^2 - 3x^2 + 10x = -25\), \(x = -5\).