Solve 18=6x+x(x-13)

$$18=6x+x(x-13)$$

Solve for x

$$x=-2$$

$$x=9$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x-13.

$$18=6x+x^{2}-13x$$

Combine 6x and -13x to get -7x.

$$18=-7x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-7x+x^{2}=18$$

Subtract 18 from both sides.

$$-7x+x^{2}-18=0$$

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.

$$x^{2}-7x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-18\right)}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2}$$

Add 49 to 72.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-\left(-7\right)±11}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±11}{2}$$

Now solve the equation x=\frac{7±11}{2} when ± is plus. Add 7 to 11.

$$x=\frac{18}{2}$$

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{7±11}{2} when ± is minus. Subtract 11 from 7.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=9 x=-2$$

Steps for Completing the Square

Use the distributive property to multiply x by x-13.

$$18=6x+x^{2}-13x$$

Combine 6x and -13x to get -7x.

$$18=-7x+x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-7x+x^{2}=18$$

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.

$$x^{2}-7x=18$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=18+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=18+\frac{49}{4}$$

Add 18 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{121}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{11}{2} x-\frac{7}{2}=-\frac{11}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=9 x=-2$$