Solve 10x=x(x+1)/2

$$10x=x \frac{ (x+1) }{ 2 }$$

Solve for x

$$x=19$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$20x=x\left(x+1\right)$$

Use the distributive property to multiply x by x+1.

$$20x=x^{2}+x$$

Subtract x^{2} from both sides.

$$20x-x^{2}=x$$

Subtract x from both sides.

$$20x-x^{2}-x=0$$

Combine 20x and -x to get 19x.

$$19x-x^{2}=0$$

Factor out x.

$$x\left(19-x\right)=0$$

To find equation solutions, solve x=0 and 19-x=0.

$$x=0 x=19$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$20x=x\left(x+1\right)$$

Use the distributive property to multiply x by x+1.

$$20x=x^{2}+x$$

Subtract x^{2} from both sides.

$$20x-x^{2}=x$$

Subtract x from both sides.

$$20x-x^{2}-x=0$$

Combine 20x and -x to get 19x.

$$19x-x^{2}=0$$

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.

$$-x^{2}+19x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 19 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}}}{2\left(-1\right)}$$

Take the square root of 19^{2}.

$$x=\frac{-19±19}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-19±19}{-2}$$

Now solve the equation x=\frac{-19±19}{-2} when ± is plus. Add -19 to 19.

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

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-19±19}{-2} when ± is minus. Subtract 19 from -19.

$$x=-\frac{38}{-2}$$

Divide -38 by -2.

$$x=19$$

The equation is now solved.

$$x=0 x=19$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$20x=x\left(x+1\right)$$

Use the distributive property to multiply x by x+1.

$$20x=x^{2}+x$$

Subtract x^{2} from both sides.

$$20x-x^{2}=x$$

Subtract x from both sides.

$$20x-x^{2}-x=0$$

Combine 20x and -x to get 19x.

$$19x-x^{2}=0$$

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.

$$-x^{2}+19x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+19x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{19}{-1}x=\frac{0}{-1}$$

Divide 19 by -1.

$$x^{2}-19x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-19x=0$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=\frac{361}{4}$$

Factor x^{2}-19x+\frac{361}{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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{19}{2} x-\frac{19}{2}=-\frac{19}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=19 x=0$$