Solve x^2-132x+468=0

$${ x }^{ 2 } -132x+468 = 0$$

Solve for x

$$x=36\sqrt{3}+66\approx 128.353829072$$

$$x=66-36\sqrt{3}\approx 3.646170928$$

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.

$$x^{2}-132x+468=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -132 for b, and 468 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-132\right)±\sqrt{\left(-132\right)^{2}-4\times 468}}{2}$$

Square -132.

$$x=\frac{-\left(-132\right)±\sqrt{17424-4\times 468}}{2}$$

Multiply -4 times 468.

$$x=\frac{-\left(-132\right)±\sqrt{17424-1872}}{2}$$

Add 17424 to -1872.

$$x=\frac{-\left(-132\right)±\sqrt{15552}}{2}$$

Take the square root of 15552.

$$x=\frac{-\left(-132\right)±72\sqrt{3}}{2}$$

The opposite of -132 is 132.

$$x=\frac{132±72\sqrt{3}}{2}$$

Now solve the equation x=\frac{132±72\sqrt{3}}{2} when ± is plus. Add 132 to 72\sqrt{3}.

$$x=\frac{72\sqrt{3}+132}{2}$$

Divide 132+72\sqrt{3} by 2.

$$x=36\sqrt{3}+66$$

Now solve the equation x=\frac{132±72\sqrt{3}}{2} when ± is minus. Subtract 72\sqrt{3} from 132.

$$x=\frac{132-72\sqrt{3}}{2}$$

Divide 132-72\sqrt{3} by 2.

$$x=66-36\sqrt{3}$$

The equation is now solved.

$$x=36\sqrt{3}+66 x=66-36\sqrt{3}$$

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.

$$x^{2}-132x+468=0$$

Subtract 468 from both sides of the equation.

$$x^{2}-132x+468-468=-468$$

Subtracting 468 from itself leaves 0.

$$x^{2}-132x=-468$$

Divide -132, the coefficient of the x term, by 2 to get -66. Then add the square of -66 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-132x+\left(-66\right)^{2}=-468+\left(-66\right)^{2}$$

Square -66.

$$x^{2}-132x+4356=-468+4356$$

Add -468 to 4356.

$$x^{2}-132x+4356=3888$$

Factor x^{2}-132x+4356. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-66\right)^{2}=3888$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-66\right)^{2}}=\sqrt{3888}$$

Simplify.

$$x-66=36\sqrt{3} x-66=-36\sqrt{3}$$

Add 66 to both sides of the equation.

$$x=36\sqrt{3}+66 x=66-36\sqrt{3}$$