Solve x^2-390x-39=8

$$x ^ { 2 } - 390 x - 39 = 8$$

Solve for x

$$x=2\sqrt{9518}+195\approx 390.120475604$$

$$x=195-2\sqrt{9518}\approx -0.120475604$$

Steps Using the Quadratic Formula

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.

$$x^{2}-390x-39=8$$

Subtract 8 from both sides of the equation.

$$x^{2}-390x-39-8=8-8$$

Subtracting 8 from itself leaves 0.

$$x^{2}-390x-39-8=0$$

Subtract 8 from -39.

$$x^{2}-390x-47=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -390 for b, and -47 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-390\right)±\sqrt{\left(-390\right)^{2}-4\left(-47\right)}}{2}$$

Square -390.

$$x=\frac{-\left(-390\right)±\sqrt{152100-4\left(-47\right)}}{2}$$

Multiply -4 times -47.

$$x=\frac{-\left(-390\right)±\sqrt{152100+188}}{2}$$

Add 152100 to 188.

$$x=\frac{-\left(-390\right)±\sqrt{152288}}{2}$$

Take the square root of 152288.

$$x=\frac{-\left(-390\right)±4\sqrt{9518}}{2}$$

The opposite of -390 is 390.

$$x=\frac{390±4\sqrt{9518}}{2}$$

Now solve the equation x=\frac{390±4\sqrt{9518}}{2} when ± is plus. Add 390 to 4\sqrt{9518}.

$$x=\frac{4\sqrt{9518}+390}{2}$$

Divide 390+4\sqrt{9518} by 2.

$$x=2\sqrt{9518}+195$$

Now solve the equation x=\frac{390±4\sqrt{9518}}{2} when ± is minus. Subtract 4\sqrt{9518} from 390.

$$x=\frac{390-4\sqrt{9518}}{2}$$

Divide 390-4\sqrt{9518} by 2.

$$x=195-2\sqrt{9518}$$

The equation is now solved.

$$x=2\sqrt{9518}+195 x=195-2\sqrt{9518}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-390x-39=8$$

Add 39 to both sides of the equation.

$$x^{2}-390x-39-\left(-39\right)=8-\left(-39\right)$$

Subtracting -39 from itself leaves 0.

$$x^{2}-390x=8-\left(-39\right)$$

Subtract -39 from 8.

$$x^{2}-390x=47$$

Divide -390, the coefficient of the x term, by 2 to get -195. Then add the square of -195 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-390x+\left(-195\right)^{2}=47+\left(-195\right)^{2}$$

Square -195.

$$x^{2}-390x+38025=47+38025$$

Add 47 to 38025.

$$x^{2}-390x+38025=38072$$

Factor x^{2}-390x+38025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-195\right)^{2}=38072$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-195\right)^{2}}=\sqrt{38072}$$

Simplify.

$$x-195=2\sqrt{9518} x-195=-2\sqrt{9518}$$

Add 195 to both sides of the equation.

$$x=2\sqrt{9518}+195 x=195-2\sqrt{9518}$$