Question

question
solve the equation $2x^2 + 9x + 12 = x^2$ to the nearest tenth.

Explanation:

Step1: Simplify the equation

Subtract \(x^2\) from both sides of the equation \(2x^2 + 9x + 12 = x^2\) to get a quadratic equation in standard form.
\(2x^2 - x^2 + 9x + 12 = 0\)
\(x^2 + 9x + 12 = 0\)

Step2: Identify coefficients for quadratic formula

For a quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = 9\), \(c = 12\). The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)

Step3: Calculate the discriminant

The discriminant \(D=b^2 - 4ac=(9)^2-4\times1\times12 = 81 - 48=33\)

Step4: Find the roots

Substitute \(a = 1\), \(b = 9\), \(D = 33\) into the quadratic formula:
\(x=\frac{-9\pm\sqrt{33}}{2\times1}=\frac{-9\pm5.7446}{2}\)

First root: \(x_1=\frac{-9 + 5.7446}{2}=\frac{-3.2554}{2}=- 1.6277\approx - 1.6\) (to the nearest tenth)
Second root: \(x_2=\frac{-9 - 5.7446}{2}=\frac{-14.7446}{2}=-7.3723\approx - 7.4\) (to the nearest tenth)

Answer:

The solutions are \(x\approx - 1.6\) and \(x\approx - 7.4\)