Question

Solve the equation 5, x, squared, minus, 6, x, minus, 11, equals, x, squared, minus, 9, x5x
2
−6x−11=x
2
−9x to the nearest tenth.

Answer 1

To solve the equation \( 5x^2 - 6x - 11 = x^2 - 9 \), we first simplify both sides.

1. Rearranging the equation gives:

\[
5x^2 - 6x - 11 - x^2 + 9 = 0.
\]

2. Combine like terms:

\[
(5x^2 - x^2) - 6x + (9 - 11) = 0,
\]
\[
4x^2 - 6x - 2 = 0.
\]

3. Next, we can simplify the equation by dividing all terms by 2:

\[
2x^2 - 3x - 1 = 0.
\]

Now we can use the quadratic formula to solve for \( x \), where \( a = 2 \), \( b = -3 \), and \( c = -1 \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\]

Substituting in our values:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2},
\]
\[
x = \frac{3 \pm \sqrt{9 + 8}}{4},
\]
\[
x = \frac{3 \pm \sqrt{17}}{4}.
\]

Now we can find the two potential solutions:

1. Using the plus sign:

\[
x_1 = \frac{3 + \sqrt{17}}{4} \approx \frac{3 + 4.1231}{4} \approx \frac{7.1231}{4} \approx 1.7808.
\]

2. Using the minus sign:

\[
x_2 = \frac{3 - \sqrt{17}}{4} \approx \frac{3 - 4.1231}{4} \approx \frac{-1.1231}{4} \approx -0.2808.
\]

Now rounding to the nearest tenth:

- For \( x_1 \approx 1.8 \).
- For \( x_2 \approx -0.3 \).

Thus, the solutions to the equation \( 5x^2 - 6x - 11 = x^2 - 9 \) rounded to the nearest tenth are:

\[
x \approx 1.8 \text{ and } x \approx -0.3.
\]