Ask a New Question
Menu

Question 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

1 answer

To determine which of the equations and their corresponding solutions are matched correctly, let's solve each equation one by one.

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

    Rearranging gives: \[ -x^2 = 64 \] \[ x^2 = -64 \]

    This equation has no real solutions (since the square of a number cannot be negative). Therefore, the provided solution \( x = -8, 8 \) is incorrect.

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

    Rearranging gives: \[ -x^2 - x = 70 \] \[ x^2 + x + 70 = 0\]

    Here, we can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\): \[a = 1, b = 1, c = 70\] \[b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 70 = 1 - 280 = -279\]

    Again, this equation has no real solutions. Thus, the provided solution \( x = -10, 7 \) is also incorrect.

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

    Rearranging gives: \[ x^2 + 10x + 25 = 0 \] This can be factored as: \[(x + 5)^2 = 0\] Thus, \(x = -5\) (a double root).

    The provided solution \(x = -5\) is correct.

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

    Rearranging gives: \[x^2 + 6x - 112 = 0\]

    Applying the quadratic formula: \[a = 1, b = 6, c = -112\] \[b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot (-112) = 36 + 448 = 484\] \[\sqrt{484} = 22\] Thus: \[x = \frac{-6 \pm 22}{2}\] This gives us: \[x = \frac{16}{2} = 8, \quad x = \frac{-28}{2} = -14\]

    Therefore, the solution \(x = -8, 14\) is incorrect.

In summary, the correctly matched equation and solution are:

  • \(4x^2 - 3x^2 + 10x = -25\), \(x = -5\)

Therefore, the correct response is: 4x2−3x2+10x=−25 , x=−5