Solve 46x=x^2

$$46 x = x ^ { 2 }$$

Solve for x

$$x=46$$

$$x=0$$

Steps Using Factoring

Subtract x^{2} from both sides.

$$46x-x^{2}=0$$

Factor out x.

$$x\left(46-x\right)=0$$

To find equation solutions, solve x=0 and 46-x=0.

$$x=0 x=46$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$46x-x^{2}=0$$

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.

$$-x^{2}+46x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 46 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46±\sqrt{46^{2}}}{2\left(-1\right)}$$

Take the square root of 46^{2}.

$$x=\frac{-46±46}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-46±46}{-2}$$

Now solve the equation x=\frac{-46±46}{-2} when ± is plus. Add -46 to 46.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-46±46}{-2} when ± is minus. Subtract 46 from -46.

$$x=-\frac{92}{-2}$$

Divide -92 by -2.

$$x=46$$

The equation is now solved.

$$x=0 x=46$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$46x-x^{2}=0$$

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.

$$-x^{2}+46x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+46x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{46}{-1}x=\frac{0}{-1}$$

Divide 46 by -1.

$$x^{2}-46x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-46x=0$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=529$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=529$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{529}$$

Simplify.

$$x-23=23 x-23=-23$$

Add 23 to both sides of the equation.

$$x=46 x=0$$