Solve x^2-288x+1136=0

$${ x }^{ 2 } -288x+1136=0$$

Solve for x

$$x=4$$

$$x=284$$

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.

$$x^{2}-288x+1136=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -288 for b, and 1136 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-288\right)±\sqrt{\left(-288\right)^{2}-4\times 1136}}{2}$$

Square -288.

$$x=\frac{-\left(-288\right)±\sqrt{82944-4\times 1136}}{2}$$

Multiply -4 times 1136.

$$x=\frac{-\left(-288\right)±\sqrt{82944-4544}}{2}$$

Add 82944 to -4544.

$$x=\frac{-\left(-288\right)±\sqrt{78400}}{2}$$

Take the square root of 78400.

$$x=\frac{-\left(-288\right)±280}{2}$$

The opposite of -288 is 288.

$$x=\frac{288±280}{2}$$

Now solve the equation x=\frac{288±280}{2} when ± is plus. Add 288 to 280.

$$x=\frac{568}{2}$$

Divide 568 by 2.

$$x=284$$

Now solve the equation x=\frac{288±280}{2} when ± is minus. Subtract 280 from 288.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

The equation is now solved.

$$x=284 x=4$$

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.

$$x^{2}-288x+1136=0$$

Subtract 1136 from both sides of the equation.

$$x^{2}-288x+1136-1136=-1136$$

Subtracting 1136 from itself leaves 0.

$$x^{2}-288x=-1136$$

Divide -288, the coefficient of the x term, by 2 to get -144. Then add the square of -144 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-288x+\left(-144\right)^{2}=-1136+\left(-144\right)^{2}$$

Square -144.

$$x^{2}-288x+20736=-1136+20736$$

Add -1136 to 20736.

$$x^{2}-288x+20736=19600$$

Factor x^{2}-288x+20736. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-144\right)^{2}=19600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-144\right)^{2}}=\sqrt{19600}$$

Simplify.

$$x-144=140 x-144=-140$$

Add 144 to both sides of the equation.

$$x=284 x=4$$