Question

Solve the equation 4, x, squared, plus, 9, x, minus, 11, equals, 2, x, squared, plus, 10, x4x
2
+9x−11=2x
2
+10x to the nearest tenth.

Answer 1

To solve the equation \( 4x^2 + 9x - 11 = 2x^2 + 10x \), we first rearrange all terms to one side of the equation.

Starting with the original equation:

\[
4x^2 + 9x - 11 = 2x^2 + 10x
\]

Subtract \( 2x^2 \) and \( 10x \) from both sides:

\[
4x^2 - 2x^2 + 9x - 10x - 11 = 0
\]

This simplifies to:

\[
2x^2 - x - 11 = 0
\]

Now, we can use the quadratic formula to solve for \( x \). The quadratic formula is given by:

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

In our equation \( 2x^2 - x - 11 = 0\), we can identify \( a = 2 \), \( b = -1 \), and \( c = -11 \).

First, we calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (-1)^2 - 4(2)(-11)
\]
\[
= 1 + 88
\]
\[
= 89
\]

Now we can apply the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{89}}{2(2)}
\]
\[
= \frac{1 \pm \sqrt{89}}{4}
\]

Next, we calculate \( \sqrt{89} \), which is approximately \( 9.434 \).

So, we have:

\[
x = \frac{1 + 9.434}{4} \quad \text{and} \quad x = \frac{1 - 9.434}{4}
\]

Calculating the first solution:

\[
x \approx \frac{10.434}{4} \approx 2.6085
\]

Calculating the second solution:

\[
x \approx \frac{-8.434}{4} \approx -2.1085
\]

Rounding both answers to the nearest tenth, we find:

- \( x \approx 2.6 \)
- \( x \approx -2.1 \)

Therefore, the solutions to the equation are:

\[
x \approx 2.6 \quad \text{and} \quad x \approx -2.1
\]