Solve the equation minus, x, squared, plus, 9, x, minus, 26, equals, minus, 2, x, squared, plus, 24−x 2 +9x−26=−2x 2 +24 to the nearest tenth.

1. Break Down the Problem

We need to solve the equation: $-x^2 + 9x - 26 = -2x^2 + 24$

2. Relevant Concepts

First, we will rearrange the equation and combine like terms to simplify it.

3. Analysis and Detail

1. Move all terms to one side of the equation: $-x^2 + 9x - 26 + 2x^2 - 24 = 0$ This can be rewritten as: $(2x^2 - x^2) + 9x + (-26 - 24) = 0$

2. Combine like terms: $x^2 + 9x - 50 = 0$

3. We will use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 9$, and $c = -50$.

4. Calculate the discriminant ($b^2 - 4ac$): $b^2 - 4ac = 9^2 - 4(1)(-50) = 81 + 200 = 281$

5. Plug into the quadratic formula: $x = \frac{-9 \pm \sqrt{281}}{2}$ Approximating $\sqrt{281}$: $\sqrt{281} \approx 16.76$

6. Calculate the values for $x$: $x_1 = \frac{-9 + 16.76}{2} \approx \frac{7.76}{2} \approx 3.88$ $x_2 = \frac{-9 - 16.76}{2} \approx \frac{-25.76}{2} \approx -12.88$

4. Verify and Summarize

Both values can be approximated to the nearest tenth:

• $x_1 \approx 3.9$
• $x_2 \approx -12.9$

Final Answer

The solutions to the equation are approximately: $x \approx 3.9 \quad \text{and} \quad x \approx -12.9$