Solve 00=x(1+13-2x)

$$00 = x ( 1 + 13 - 2 x )$$

Solve for x

$$x=7$$

$$x=0$$

Steps Using Factoring

Multiply 0 and 0 to get 0.

$$0=x\left(1+13-2x\right)$$

Add 1 and 13 to get 14.

$$0=x\left(14-2x\right)$$

Use the distributive property to multiply x by 14-2x.

$$0=14x-2x^{2}$$

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

$$14x-2x^{2}=0$$

Factor out x.

$$x\left(14-2x\right)=0$$

To find equation solutions, solve x=0 and 14-2x=0.

$$x=0 x=7$$

Steps Using the Quadratic Formula

Multiply 0 and 0 to get 0.

$$0=x\left(1+13-2x\right)$$

Add 1 and 13 to get 14.

$$0=x\left(14-2x\right)$$

Use the distributive property to multiply x by 14-2x.

$$0=14x-2x^{2}$$

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

$$14x-2x^{2}=0$$

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.

$$-2x^{2}+14x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 14 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}}}{2\left(-2\right)}$$

Take the square root of 14^{2}.

$$x=\frac{-14±14}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-14±14}{-4}$$

Now solve the equation x=\frac{-14±14}{-4} when ± is plus. Add -14 to 14.

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

Now solve the equation x=\frac{-14±14}{-4} when ± is minus. Subtract 14 from -14.

$$x=-\frac{28}{-4}$$

Divide -28 by -4.

$$x=7$$

The equation is now solved.

$$x=0 x=7$$

Steps for Completing the Square

Multiply 0 and 0 to get 0.

$$0=x\left(1+13-2x\right)$$

Add 1 and 13 to get 14.

$$0=x\left(14-2x\right)$$

Use the distributive property to multiply x by 14-2x.

$$0=14x-2x^{2}$$

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

$$14x-2x^{2}=0$$

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.

$$-2x^{2}+14x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}+14x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{14}{-2}x=\frac{0}{-2}$$

Divide 14 by -2.

$$x^{2}-7x=\frac{0}{-2}$$

Divide 0 by -2.

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

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.

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

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

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$x=7 x=0$$