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

$$(x+1)(x+2)=(2x-1)(2x-10)$$

Solve for x

$$x=8$$

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using the Quadratic Formula

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

$$x^{2}+3x+2=\left(2x-1\right)\left(2x-10\right)$$

Use the distributive property to multiply 2x-1 by 2x-10 and combine like terms.

$$x^{2}+3x+2=4x^{2}-22x+10$$

Subtract 4x^{2} from both sides.

$$x^{2}+3x+2-4x^{2}=-22x+10$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}+3x+2=-22x+10$$

Add 22x to both sides.

$$-3x^{2}+3x+2+22x=10$$

Combine 3x and 22x to get 25x.

$$-3x^{2}+25x+2=10$$

Subtract 10 from both sides.

$$-3x^{2}+25x+2-10=0$$

Subtract 10 from 2 to get -8.

$$-3x^{2}+25x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 25 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-3\right)\left(-8\right)}}{2\left(-3\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-3\right)\left(-8\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-25±\sqrt{625+12\left(-8\right)}}{2\left(-3\right)}$$

Multiply 12 times -8.

$$x=\frac{-25±\sqrt{625-96}}{2\left(-3\right)}$$

Add 625 to -96.

$$x=\frac{-25±\sqrt{529}}{2\left(-3\right)}$$

Take the square root of 529.

$$x=\frac{-25±23}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-25±23}{-6}$$

Now solve the equation x=\frac{-25±23}{-6} when ± is plus. Add -25 to 23.

$$x=-\frac{2}{-6}$$

Reduce the fraction \frac{-2}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-25±23}{-6} when ± is minus. Subtract 23 from -25.

$$x=-\frac{48}{-6}$$

Divide -48 by -6.

$$x=8$$

The equation is now solved.

$$x=\frac{1}{3} x=8$$

Steps for Completing the Square

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

$$x^{2}+3x+2=\left(2x-1\right)\left(2x-10\right)$$

Use the distributive property to multiply 2x-1 by 2x-10 and combine like terms.

$$x^{2}+3x+2=4x^{2}-22x+10$$

Subtract 4x^{2} from both sides.

$$x^{2}+3x+2-4x^{2}=-22x+10$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}+3x+2=-22x+10$$

Add 22x to both sides.

$$-3x^{2}+3x+2+22x=10$$

Combine 3x and 22x to get 25x.

$$-3x^{2}+25x+2=10$$

Subtract 2 from both sides.

$$-3x^{2}+25x=10-2$$

Subtract 2 from 10 to get 8.

$$-3x^{2}+25x=8$$

Divide both sides by -3.

$$\frac{-3x^{2}+25x}{-3}=\frac{8}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{25}{-3}x=\frac{8}{-3}$$

Divide 25 by -3.

$$x^{2}-\frac{25}{3}x=\frac{8}{-3}$$

Divide 8 by -3.

$$x^{2}-\frac{25}{3}x=-\frac{8}{3}$$

Divide -\frac{25}{3}, the coefficient of the x term, by 2 to get -\frac{25}{6}. Then add the square of -\frac{25}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{3}x+\left(-\frac{25}{6}\right)^{2}=-\frac{8}{3}+\left(-\frac{25}{6}\right)^{2}$$

Square -\frac{25}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{3}x+\frac{625}{36}=-\frac{8}{3}+\frac{625}{36}$$

Add -\frac{8}{3} to \frac{625}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{3}x+\frac{625}{36}=\frac{529}{36}$$

Factor x^{2}-\frac{25}{3}x+\frac{625}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{6}\right)^{2}=\frac{529}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{6}\right)^{2}}=\sqrt{\frac{529}{36}}$$

Simplify.

$$x-\frac{25}{6}=\frac{23}{6} x-\frac{25}{6}=-\frac{23}{6}$$

Add \frac{25}{6} to both sides of the equation.

$$x=8 x=\frac{1}{3}$$