Which of the following equations and solutions is matched correctly?(1 point)Responses−x2−64=0, x=−8, 8−x2−64=0, x=−8, 84x2−3x2+10x=−25, x=−54x2−3x2+10x=−25, x=−5−x2−x−70=0, x=−10, 7−x2−x−70=0, x=−10, 7x2+6x=112, x=−8, 14

Answer 1

Let's match each equation with the corresponding solutions correctly by solving the equations one by one.

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

   Rearranging, we get: \[-x^2 = 64 \implies x^2 = -64\] This has no real solutions (the solutions are imaginary).

2. Equation: \(-x^2 - 64 = 0\) (the same as the first one)

   This has the same conclusion as above.

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

   Simplifying gives: \[x^2 + 10x + 25 = 0\] Factoring (or using the quadratic formula): \[(x + 5)^2 = 0\] This leads to a double root \(x = -5\).

4. Equation: \(4x^2 - 3x^2 + 10x = -25\) (the same as the previous one)

   This has the same conclusion as above.

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

   Rearranging gives: \[-x^2 - x = 70\] \[x^2 + x + 70 = 0\] The discriminant is \(b^2 - 4ac = 1 - 280 < 0\), indicating no real solutions.

6. Equation: \(7 - x^2 - x - 70 = 0\)

   Rearranging gives: \[-x^2 - x - 63 = 0\] \[x^2 + x + 63 = 0\] The discriminant here is \(b^2 - 4ac = 1 - 252 < 0\), again indicating no real solutions.

7. Equation: \(7x^2 + 6x = 112\)

   Rearranging gives: \[7x^2 + 6x - 112 = 0\] Using the quadratic formula: The discriminant will be \(b^2 - 4ac = 6^2 - 4(7)(-112)\): \[= 36 + 3136 = 3172\] Solving using the quadratic formula: \[x = \frac{-6 \pm \sqrt{3172}}{14}\] This will yield two real solutions, but calculating exact values is complex without a calculator. Let's check potential values \(x = -8\) and \(x = 14\).

Now, comparing the equations with provided solutions:

• For equation 3: \(4x^2 - 3x^2 + 10x = -25\), matches with \(x = -5\).
• For equation 5 and 6: they both do not have correct solutions listed, as they give imaginary roots.
• For equation 7: \(7x^2 + 6x = 112\) does not match \(x=-8\) or \(x=14\) straightforwardly but needs to be numerically checked.

Matched Correctly:

The only match we definitely have is the third equation \(4x^2 - 3x^2 + 10x = -25\) with its solution \(x = -5\). All others do not have their solutions accurately stated in provided options.

Conclusively, the correctly matched pair is: \(4x^2 - 3x^2 + 10x = -25\), \(x = -5\).