Solve (x-1)(x+2)+3x=4(x-2)-(x-12)

$$( x - 1 ) ( x + 2 ) + 3 x = 4 ( x - 2 ) - ( x - 12 )$$

Solve for x

$$x=-3$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$x^{2}+x-2+3x=4\left(x-2\right)-\left(x-12\right)$$

Combine x and 3x to get 4x.

$$x^{2}+4x-2=4\left(x-2\right)-\left(x-12\right)$$

Use the distributive property to multiply 4 by x-2.

$$x^{2}+4x-2=4x-8-\left(x-12\right)$$

To find the opposite of x-12, find the opposite of each term.

$$x^{2}+4x-2=4x-8-x+12$$

Combine 4x and -x to get 3x.

$$x^{2}+4x-2=3x-8+12$$

Add -8 and 12 to get 4.

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

Subtract 3x from both sides.

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

Combine 4x and -3x to get x.

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

Subtract 4 from both sides.

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

Subtract 4 from -2 to get -6.

$$x^{2}+x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-6\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-1±\sqrt{1+24}}{2}$$

Add 1 to 24.

$$x=\frac{-1±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-1±5}{2}$$

Now solve the equation x=\frac{-1±5}{2} when ± is plus. Add -1 to 5.

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

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-1±5}{2} when ± is minus. Subtract 5 from -1.

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

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=2 x=-3$$

Steps for Completing the Square

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

$$x^{2}+x-2+3x=4\left(x-2\right)-\left(x-12\right)$$

Combine x and 3x to get 4x.

$$x^{2}+4x-2=4\left(x-2\right)-\left(x-12\right)$$

Use the distributive property to multiply 4 by x-2.

$$x^{2}+4x-2=4x-8-\left(x-12\right)$$

To find the opposite of x-12, find the opposite of each term.

$$x^{2}+4x-2=4x-8-x+12$$

Combine 4x and -x to get 3x.

$$x^{2}+4x-2=3x-8+12$$

Add -8 and 12 to get 4.

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

Subtract 3x from both sides.

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

Combine 4x and -3x to get x.

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

Add 2 to both sides.

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

Add 4 and 2 to get 6.

$$x^{2}+x=6$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=6+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=6+\frac{1}{4}$$

Add 6 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{25}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

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

Subtract \frac{1}{2} from both sides of the equation.

$$x=2 x=-3$$