Solve x^2+529x-252=0

$${ x }^{ 2 } +529x-252=0$$

Solve for x

$$x=\frac{\sqrt{280849}-529}{2}\approx 0.475942304$$

$$x=\frac{-\sqrt{280849}-529}{2}\approx -529.475942304$$

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.

$$x^{2}+529x-252=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 529 for b, and -252 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-529±\sqrt{529^{2}-4\left(-252\right)}}{2}$$

Square 529.

$$x=\frac{-529±\sqrt{279841-4\left(-252\right)}}{2}$$

Multiply -4 times -252.

$$x=\frac{-529±\sqrt{279841+1008}}{2}$$

Add 279841 to 1008.

$$x=\frac{-529±\sqrt{280849}}{2}$$

Now solve the equation x=\frac{-529±\sqrt{280849}}{2} when ± is plus. Add -529 to \sqrt{280849}.

$$x=\frac{\sqrt{280849}-529}{2}$$

Now solve the equation x=\frac{-529±\sqrt{280849}}{2} when ± is minus. Subtract \sqrt{280849} from -529.

$$x=\frac{-\sqrt{280849}-529}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{280849}-529}{2} x=\frac{-\sqrt{280849}-529}{2}$$

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.

$$x^{2}+529x-252=0$$

Add 252 to both sides of the equation.

$$x^{2}+529x-252-\left(-252\right)=-\left(-252\right)$$

Subtracting -252 from itself leaves 0.

$$x^{2}+529x=-\left(-252\right)$$

Subtract -252 from 0.

$$x^{2}+529x=252$$

Divide 529, the coefficient of the x term, by 2 to get \frac{529}{2}. Then add the square of \frac{529}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+529x+\left(\frac{529}{2}\right)^{2}=252+\left(\frac{529}{2}\right)^{2}$$

Square \frac{529}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+529x+\frac{279841}{4}=252+\frac{279841}{4}$$

Add 252 to \frac{279841}{4}.

$$x^{2}+529x+\frac{279841}{4}=\frac{280849}{4}$$

Factor x^{2}+529x+\frac{279841}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{529}{2}\right)^{2}=\frac{280849}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{529}{2}\right)^{2}}=\sqrt{\frac{280849}{4}}$$

Simplify.

$$x+\frac{529}{2}=\frac{\sqrt{280849}}{2} x+\frac{529}{2}=-\frac{\sqrt{280849}}{2}$$

Subtract \frac{529}{2} from both sides of the equation.

$$x=\frac{\sqrt{280849}-529}{2} x=\frac{-\sqrt{280849}-529}{2}$$