Solve 4x^2-9x-6

$$4 { x }^{ 2 } -9x-6$$

Factor

$$4\left(x-\frac{9-\sqrt{177}}{8}\right)\left(x-\frac{\sqrt{177}+9}{8}\right)$$

Steps Using the Quadratic Formula

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.

$$4x^{2}-9x-6=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\left(-6\right)}}{2\times 4}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\left(-6\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\left(-6\right)}}{2\times 4}$$

Multiply -16 times -6.

$$x=\frac{-\left(-9\right)±\sqrt{81+96}}{2\times 4}$$

Add 81 to 96.

$$x=\frac{-\left(-9\right)±\sqrt{177}}{2\times 4}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{177}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{9±\sqrt{177}}{8}$$

Now solve the equation x=\frac{9±\sqrt{177}}{8} when ± is plus. Add 9 to \sqrt{177}.

$$x=\frac{\sqrt{177}+9}{8}$$

Now solve the equation x=\frac{9±\sqrt{177}}{8} when ± is minus. Subtract \sqrt{177} from 9.

$$x=\frac{9-\sqrt{177}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9+\sqrt{177}}{8} for x_{1} and \frac{9-\sqrt{177}}{8} for x_{2}.

$$4x^{2}-9x-6=4\left(x-\frac{\sqrt{177}+9}{8}\right)\left(x-\frac{9-\sqrt{177}}{8}\right)$$

Evaluate

$$4x^{2}-9x-6$$