Solve 3x^2-6x-69

$$3 { x }^{ 2 } -6x-69$$

Factor

$$3\left(x-\left(1-2\sqrt{6}\right)\right)\left(x-\left(2\sqrt{6}+1\right)\right)$$

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.

$$3x^{2}-6x-69=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-69\right)}}{2\times 3}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-69\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\left(-69\right)}}{2\times 3}$$

Multiply -12 times -69.

$$x=\frac{-\left(-6\right)±\sqrt{36+828}}{2\times 3}$$

Add 36 to 828.

$$x=\frac{-\left(-6\right)±\sqrt{864}}{2\times 3}$$

Take the square root of 864.

$$x=\frac{-\left(-6\right)±12\sqrt{6}}{2\times 3}$$

The opposite of -6 is 6.

$$x=\frac{6±12\sqrt{6}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±12\sqrt{6}}{6}$$

Now solve the equation x=\frac{6±12\sqrt{6}}{6} when ± is plus. Add 6 to 12\sqrt{6}.

$$x=\frac{12\sqrt{6}+6}{6}$$

Divide 6+12\sqrt{6} by 6.

$$x=2\sqrt{6}+1$$

Now solve the equation x=\frac{6±12\sqrt{6}}{6} when ± is minus. Subtract 12\sqrt{6} from 6.

$$x=\frac{6-12\sqrt{6}}{6}$$

Divide 6-12\sqrt{6} by 6.

$$x=1-2\sqrt{6}$$

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

$$3x^{2}-6x-69=3\left(x-\left(2\sqrt{6}+1\right)\right)\left(x-\left(1-2\sqrt{6}\right)\right)$$

Evaluate

$$3\left(x^{2}-2x-23\right)$$