Solve 452x/45=x^2

$$\frac{ 452x }{ 45 } = { x }^{ 2 }$$

Solve for x

$$x = \frac{452}{45} = 10\frac{2}{45} \approx 10.044444444$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 45.

$$452x=45x^{2}$$

Subtract 45x^{2} from both sides.

$$452x-45x^{2}=0$$

Factor out x.

$$x\left(452-45x\right)=0$$

To find equation solutions, solve x=0 and 452-45x=0.

$$x=0 x=\frac{452}{45}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 45.

$$452x=45x^{2}$$

Subtract 45x^{2} from both sides.

$$452x-45x^{2}=0$$

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.

$$-45x^{2}+452x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -45 for a, 452 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-452±\sqrt{452^{2}}}{2\left(-45\right)}$$

Take the square root of 452^{2}.

$$x=\frac{-452±452}{2\left(-45\right)}$$

Multiply 2 times -45.

$$x=\frac{-452±452}{-90}$$

Now solve the equation x=\frac{-452±452}{-90} when ± is plus. Add -452 to 452.

$$x=\frac{0}{-90}$$

Divide 0 by -90.

$$x=0$$

Now solve the equation x=\frac{-452±452}{-90} when ± is minus. Subtract 452 from -452.

$$x=-\frac{904}{-90}$$

Reduce the fraction \frac{-904}{-90} to lowest terms by extracting and canceling out 2.

$$x=\frac{452}{45}$$

The equation is now solved.

$$x=0 x=\frac{452}{45}$$

Steps for Completing the Square

Multiply both sides of the equation by 45.

$$452x=45x^{2}$$

Subtract 45x^{2} from both sides.

$$452x-45x^{2}=0$$

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.

$$-45x^{2}+452x=0$$

Divide both sides by -45.

$$\frac{-45x^{2}+452x}{-45}=\frac{0}{-45}$$

Dividing by -45 undoes the multiplication by -45.

$$x^{2}+\frac{452}{-45}x=\frac{0}{-45}$$

Divide 452 by -45.

$$x^{2}-\frac{452}{45}x=\frac{0}{-45}$$

Divide 0 by -45.

$$x^{2}-\frac{452}{45}x=0$$

Divide -\frac{452}{45}, the coefficient of the x term, by 2 to get -\frac{226}{45}. Then add the square of -\frac{226}{45} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{452}{45}x+\left(-\frac{226}{45}\right)^{2}=\left(-\frac{226}{45}\right)^{2}$$

Square -\frac{226}{45} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{452}{45}x+\frac{51076}{2025}=\frac{51076}{2025}$$

Factor x^{2}-\frac{452}{45}x+\frac{51076}{2025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{226}{45}\right)^{2}=\frac{51076}{2025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{226}{45}\right)^{2}}=\sqrt{\frac{51076}{2025}}$$

Simplify.

$$x-\frac{226}{45}=\frac{226}{45} x-\frac{226}{45}=-\frac{226}{45}$$

Add \frac{226}{45} to both sides of the equation.

$$x=\frac{452}{45} x=0$$