Solve (x-1)(x+2)-(2x-3)(x+4)-x+14=0

$$( x - 1 ) ( x + 2 ) - ( 2 x - 3 ) ( x + 4 ) - x + 14 = 0$$

Solve for x

$$x=-8$$

$$x=3$$

Steps Using Factoring By Grouping

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

$$x^{2}+x-2-\left(2x-3\right)\left(x+4\right)-x+14=0$$

Use the distributive property to multiply 2x-3 by x+4 and combine like terms.

$$x^{2}+x-2-\left(2x^{2}+5x-12\right)-x+14=0$$

To find the opposite of 2x^{2}+5x-12, find the opposite of each term.

$$x^{2}+x-2-2x^{2}-5x+12-x+14=0$$

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

$$-x^{2}+x-2-5x+12-x+14=0$$

Combine x and -5x to get -4x.

$$-x^{2}-4x-2+12-x+14=0$$

Add -2 and 12 to get 10.

$$-x^{2}-4x+10-x+14=0$$

Combine -4x and -x to get -5x.

$$-x^{2}-5x+10+14=0$$

Add 10 and 14 to get 24.

$$-x^{2}-5x+24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -5.

$$a=3 b=-8$$

Rewrite -x^{2}-5x+24 as \left(-x^{2}+3x\right)+\left(-8x+24\right).

$$\left(-x^{2}+3x\right)+\left(-8x+24\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(-x+3\right)+8\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(x+8\right)$$

To find equation solutions, solve -x+3=0 and x+8=0.

$$x=3 x=-8$$

Steps Using the Quadratic Formula

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

$$x^{2}+x-2-\left(2x-3\right)\left(x+4\right)-x+14=0$$

Use the distributive property to multiply 2x-3 by x+4 and combine like terms.

$$x^{2}+x-2-\left(2x^{2}+5x-12\right)-x+14=0$$

To find the opposite of 2x^{2}+5x-12, find the opposite of each term.

$$x^{2}+x-2-2x^{2}-5x+12-x+14=0$$

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

$$-x^{2}+x-2-5x+12-x+14=0$$

Combine x and -5x to get -4x.

$$-x^{2}-4x-2+12-x+14=0$$

Add -2 and 12 to get 10.

$$-x^{2}-4x+10-x+14=0$$

Combine -4x and -x to get -5x.

$$-x^{2}-5x+10+14=0$$

Add 10 and 14 to get 24.

$$-x^{2}-5x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -5 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-\left(-5\right)±\sqrt{25+96}}{2\left(-1\right)}$$

Add 25 to 96.

$$x=\frac{-\left(-5\right)±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-\left(-5\right)±11}{2\left(-1\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{5±11}{-2}$$

Now solve the equation x=\frac{5±11}{-2} when ± is plus. Add 5 to 11.

$$x=\frac{16}{-2}$$

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{5±11}{-2} when ± is minus. Subtract 11 from 5.

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

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=-8 x=3$$

Steps for Completing the Square

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

$$x^{2}+x-2-\left(2x-3\right)\left(x+4\right)-x+14=0$$

Use the distributive property to multiply 2x-3 by x+4 and combine like terms.

$$x^{2}+x-2-\left(2x^{2}+5x-12\right)-x+14=0$$

To find the opposite of 2x^{2}+5x-12, find the opposite of each term.

$$x^{2}+x-2-2x^{2}-5x+12-x+14=0$$

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

$$-x^{2}+x-2-5x+12-x+14=0$$

Combine x and -5x to get -4x.

$$-x^{2}-4x-2+12-x+14=0$$

Add -2 and 12 to get 10.

$$-x^{2}-4x+10-x+14=0$$

Combine -4x and -x to get -5x.

$$-x^{2}-5x+10+14=0$$

Add 10 and 14 to get 24.

$$-x^{2}-5x+24=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$-x^{2}-5x=-24$$

Divide both sides by -1.

$$\frac{-x^{2}-5x}{-1}=-\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{5}{-1}\right)x=-\frac{24}{-1}$$

Divide -5 by -1.

$$x^{2}+5x=-\frac{24}{-1}$$

Divide -24 by -1.

$$x^{2}+5x=24$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=24+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=24+\frac{25}{4}$$

Add 24 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{121}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{11}{2} x+\frac{5}{2}=-\frac{11}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=3 x=-8$$