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.

1 answer

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:

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