Solve x=(2x-5)(x+1)

$$x = ( 2 x - 5 ) ( x + 1 )$$

Solve for x

$$x=\frac{\sqrt{14}}{2}+1\approx 2.870828693$$

$$x=-\frac{\sqrt{14}}{2}+1\approx -0.870828693$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-5 by x+1 and combine like terms.

$$x=2x^{2}-3x-5$$

Subtract 2x^{2} from both sides.

$$x-2x^{2}=-3x-5$$

Add 3x to both sides.

$$x-2x^{2}+3x=-5$$

Combine x and 3x to get 4x.

$$4x-2x^{2}=-5$$

Add 5 to both sides.

$$4x-2x^{2}+5=0$$

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.

$$-2x^{2}+4x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 4 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 5}}{2\left(-2\right)}$$

Multiply 8 times 5.

$$x=\frac{-4±\sqrt{16+40}}{2\left(-2\right)}$$

Add 16 to 40.

$$x=\frac{-4±\sqrt{56}}{2\left(-2\right)}$$

Take the square root of 56.

$$x=\frac{-4±2\sqrt{14}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±2\sqrt{14}}{-4}$$

Now solve the equation x=\frac{-4±2\sqrt{14}}{-4} when ± is plus. Add -4 to 2\sqrt{14}.

$$x=\frac{2\sqrt{14}-4}{-4}$$

Divide -4+2\sqrt{14} by -4.

$$x=-\frac{\sqrt{14}}{2}+1$$

Now solve the equation x=\frac{-4±2\sqrt{14}}{-4} when ± is minus. Subtract 2\sqrt{14} from -4.

$$x=\frac{-2\sqrt{14}-4}{-4}$$

Divide -4-2\sqrt{14} by -4.

$$x=\frac{\sqrt{14}}{2}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{14}}{2}+1 x=\frac{\sqrt{14}}{2}+1$$

Steps for Completing the Square

Use the distributive property to multiply 2x-5 by x+1 and combine like terms.

$$x=2x^{2}-3x-5$$

Subtract 2x^{2} from both sides.

$$x-2x^{2}=-3x-5$$

Add 3x to both sides.

$$x-2x^{2}+3x=-5$$

Combine x and 3x to get 4x.

$$4x-2x^{2}=-5$$

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.

$$-2x^{2}+4x=-5$$

Divide both sides by -2.

$$\frac{-2x^{2}+4x}{-2}=-\frac{5}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{4}{-2}x=-\frac{5}{-2}$$

Divide 4 by -2.

$$x^{2}-2x=-\frac{5}{-2}$$

Divide -5 by -2.

$$x^{2}-2x=\frac{5}{2}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{5}{2}+1$$

Add \frac{5}{2} to 1.

$$x^{2}-2x+1=\frac{7}{2}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{7}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{7}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{14}}{2} x-1=-\frac{\sqrt{14}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{14}}{2}+1 x=-\frac{\sqrt{14}}{2}+1$$