Solve 144x=225x^2-50625

$$144x = 225 { x }^{ 2 } -50625$$

Solve for x

$$x = \frac{\sqrt{140689} + 8}{25} \approx 15.323412945$$

$$x=\frac{8-\sqrt{140689}}{25}\approx -14.683412945$$

Steps Using the Quadratic Formula

Subtract 225x^{2} from both sides.

$$144x-225x^{2}=-50625$$

Add 50625 to both sides.

$$144x-225x^{2}+50625=0$$

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.

$$-225x^{2}+144x+50625=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -225 for a, 144 for b, and 50625 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-144±\sqrt{144^{2}-4\left(-225\right)\times 50625}}{2\left(-225\right)}$$

Square 144.

$$x=\frac{-144±\sqrt{20736-4\left(-225\right)\times 50625}}{2\left(-225\right)}$$

Multiply -4 times -225.

$$x=\frac{-144±\sqrt{20736+900\times 50625}}{2\left(-225\right)}$$

Multiply 900 times 50625.

$$x=\frac{-144±\sqrt{20736+45562500}}{2\left(-225\right)}$$

Add 20736 to 45562500.

$$x=\frac{-144±\sqrt{45583236}}{2\left(-225\right)}$$

Take the square root of 45583236.

$$x=\frac{-144±18\sqrt{140689}}{2\left(-225\right)}$$

Multiply 2 times -225.

$$x=\frac{-144±18\sqrt{140689}}{-450}$$

Now solve the equation x=\frac{-144±18\sqrt{140689}}{-450} when ± is plus. Add -144 to 18\sqrt{140689}.

$$x=\frac{18\sqrt{140689}-144}{-450}$$

Divide -144+18\sqrt{140689} by -450.

$$x=\frac{8-\sqrt{140689}}{25}$$

Now solve the equation x=\frac{-144±18\sqrt{140689}}{-450} when ± is minus. Subtract 18\sqrt{140689} from -144.

$$x=\frac{-18\sqrt{140689}-144}{-450}$$

Divide -144-18\sqrt{140689} by -450.

$$x=\frac{\sqrt{140689}+8}{25}$$

The equation is now solved.

$$x=\frac{8-\sqrt{140689}}{25} x=\frac{\sqrt{140689}+8}{25}$$

Steps for Completing the Square

Subtract 225x^{2} from both sides.

$$144x-225x^{2}=-50625$$

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.

$$-225x^{2}+144x=-50625$$

Divide both sides by -225.

$$\frac{-225x^{2}+144x}{-225}=-\frac{50625}{-225}$$

Dividing by -225 undoes the multiplication by -225.

$$x^{2}+\frac{144}{-225}x=-\frac{50625}{-225}$$

Reduce the fraction \frac{144}{-225} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{16}{25}x=-\frac{50625}{-225}$$

Divide -50625 by -225.

$$x^{2}-\frac{16}{25}x=225$$

Divide -\frac{16}{25}, the coefficient of the x term, by 2 to get -\frac{8}{25}. Then add the square of -\frac{8}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{25}x+\left(-\frac{8}{25}\right)^{2}=225+\left(-\frac{8}{25}\right)^{2}$$

Square -\frac{8}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{25}x+\frac{64}{625}=225+\frac{64}{625}$$

Add 225 to \frac{64}{625}.

$$x^{2}-\frac{16}{25}x+\frac{64}{625}=\frac{140689}{625}$$

Factor x^{2}-\frac{16}{25}x+\frac{64}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{25}\right)^{2}=\frac{140689}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{25}\right)^{2}}=\sqrt{\frac{140689}{625}}$$

Simplify.

$$x-\frac{8}{25}=\frac{\sqrt{140689}}{25} x-\frac{8}{25}=-\frac{\sqrt{140689}}{25}$$

Add \frac{8}{25} to both sides of the equation.

$$x=\frac{\sqrt{140689}+8}{25} x=\frac{8-\sqrt{140689}}{25}$$