Solve x^2-28x-52

$${ x }^{ 2 } -28x-52$$

Factor

$$\left(x-\left(14-2\sqrt{62}\right)\right)\left(x-\left(2\sqrt{62}+14\right)\right)$$

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.

$$x^{2}-28x-52=0$$

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.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\left(-52\right)}}{2}$$

Square -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\left(-52\right)}}{2}$$

Multiply -4 times -52.

$$x=\frac{-\left(-28\right)±\sqrt{784+208}}{2}$$

Add 784 to 208.

$$x=\frac{-\left(-28\right)±\sqrt{992}}{2}$$

Take the square root of 992.

$$x=\frac{-\left(-28\right)±4\sqrt{62}}{2}$$

The opposite of -28 is 28.

$$x=\frac{28±4\sqrt{62}}{2}$$

Now solve the equation x=\frac{28±4\sqrt{62}}{2} when ± is plus. Add 28 to 4\sqrt{62}.

$$x=\frac{4\sqrt{62}+28}{2}$$

Divide 28+4\sqrt{62} by 2.

$$x=2\sqrt{62}+14$$

Now solve the equation x=\frac{28±4\sqrt{62}}{2} when ± is minus. Subtract 4\sqrt{62} from 28.

$$x=\frac{28-4\sqrt{62}}{2}$$

Divide 28-4\sqrt{62} by 2.

$$x=14-2\sqrt{62}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 14+2\sqrt{62} for x_{1} and 14-2\sqrt{62} for x_{2}.

$$x^{2}-28x-52=\left(x-\left(2\sqrt{62}+14\right)\right)\left(x-\left(14-2\sqrt{62}\right)\right)$$

Evaluate

$$x^{2}-28x-52$$