Solve 252x^2-205x=-13

$$252 { x }^{ 2 } -205x=-13$$

Solve for x

$$x=\frac{\sqrt{28921}+205}{504}\approx 0.744170146$$

$$x=\frac{205-\sqrt{28921}}{504}\approx 0.069321918$$

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.

$$252x^{2}-205x=-13$$

Add 13 to both sides of the equation.

$$252x^{2}-205x-\left(-13\right)=-13-\left(-13\right)$$

Subtracting -13 from itself leaves 0.

$$252x^{2}-205x-\left(-13\right)=0$$

Subtract -13 from 0.

$$252x^{2}-205x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 252 for a, -205 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-205\right)±\sqrt{\left(-205\right)^{2}-4\times 252\times 13}}{2\times 252}$$

Square -205.

$$x=\frac{-\left(-205\right)±\sqrt{42025-4\times 252\times 13}}{2\times 252}$$

Multiply -4 times 252.

$$x=\frac{-\left(-205\right)±\sqrt{42025-1008\times 13}}{2\times 252}$$

Multiply -1008 times 13.

$$x=\frac{-\left(-205\right)±\sqrt{42025-13104}}{2\times 252}$$

Add 42025 to -13104.

$$x=\frac{-\left(-205\right)±\sqrt{28921}}{2\times 252}$$

The opposite of -205 is 205.

$$x=\frac{205±\sqrt{28921}}{2\times 252}$$

Multiply 2 times 252.

$$x=\frac{205±\sqrt{28921}}{504}$$

Now solve the equation x=\frac{205±\sqrt{28921}}{504} when ± is plus. Add 205 to \sqrt{28921}.

$$x=\frac{\sqrt{28921}+205}{504}$$

Now solve the equation x=\frac{205±\sqrt{28921}}{504} when ± is minus. Subtract \sqrt{28921} from 205.

$$x=\frac{205-\sqrt{28921}}{504}$$

The equation is now solved.

$$x=\frac{\sqrt{28921}+205}{504} x=\frac{205-\sqrt{28921}}{504}$$

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.

$$252x^{2}-205x=-13$$

Divide both sides by 252.

$$\frac{252x^{2}-205x}{252}=-\frac{13}{252}$$

Dividing by 252 undoes the multiplication by 252.

$$x^{2}-\frac{205}{252}x=-\frac{13}{252}$$

Divide -\frac{205}{252}, the coefficient of the x term, by 2 to get -\frac{205}{504}. Then add the square of -\frac{205}{504} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{205}{252}x+\left(-\frac{205}{504}\right)^{2}=-\frac{13}{252}+\left(-\frac{205}{504}\right)^{2}$$

Square -\frac{205}{504} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{205}{252}x+\frac{42025}{254016}=-\frac{13}{252}+\frac{42025}{254016}$$

Add -\frac{13}{252} to \frac{42025}{254016} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{205}{252}x+\frac{42025}{254016}=\frac{28921}{254016}$$

Factor x^{2}-\frac{205}{252}x+\frac{42025}{254016}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{205}{504}\right)^{2}=\frac{28921}{254016}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{205}{504}\right)^{2}}=\sqrt{\frac{28921}{254016}}$$

Simplify.

$$x-\frac{205}{504}=\frac{\sqrt{28921}}{504} x-\frac{205}{504}=-\frac{\sqrt{28921}}{504}$$

Add \frac{205}{504} to both sides of the equation.

$$x=\frac{\sqrt{28921}+205}{504} x=\frac{205-\sqrt{28921}}{504}$$