Solve 8x^2+3-4x-5x^2-9x

$$8 { x }^{ 2 } +3-4x-5 { x }^{ 2 } -9x$$

Evaluate

$$3x^{2}-13x+3$$

Solution Steps

Combine 8x^{2} and -5x^{2} to get 3x^{2}.

$$3x^{2}+3-4x-9x$$

Combine -4x and -9x to get -13x.

$$3x^{2}+3-13x$$

Factor

$$3\left(x-\frac{13-\sqrt{133}}{6}\right)\left(x-\frac{\sqrt{133}+13}{6}\right)$$

Steps Using the Quadratic Formula

Combine 8x^{2} and -5x^{2} to get 3x^{2}.

$$factor(3x^{2}+3-4x-9x)$$

Combine -4x and -9x to get -13x.

$$factor(3x^{2}+3-13x)$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$3x^{2}-13x+3=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 3\times 3}}{2\times 3}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 3\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-12\times 3}}{2\times 3}$$

Multiply -12 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-36}}{2\times 3}$$

Add 169 to -36.

$$x=\frac{-\left(-13\right)±\sqrt{133}}{2\times 3}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{133}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{13±\sqrt{133}}{6}$$

Now solve the equation x=\frac{13±\sqrt{133}}{6} when ± is plus. Add 13 to \sqrt{133}.

$$x=\frac{\sqrt{133}+13}{6}$$

Now solve the equation x=\frac{13±\sqrt{133}}{6} when ± is minus. Subtract \sqrt{133} from 13.

$$x=\frac{13-\sqrt{133}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{133}}{6} for x_{1} and \frac{13-\sqrt{133}}{6} for x_{2}.

$$3x^{2}-13x+3=3\left(x-\frac{\sqrt{133}+13}{6}\right)\left(x-\frac{13-\sqrt{133}}{6}\right)$$