Solve (x+2)(-8x+12)+4quad=x^2

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

Solve for x

$$x=-2$$

$$x = \frac{14}{9} = 1\frac{5}{9} \approx 1.555555556$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x+2 by -8x+12 and combine like terms.

$$-8x^{2}-4x+24+4=x^{2}$$

Add 24 and 4 to get 28.

$$-8x^{2}-4x+28=x^{2}$$

Subtract x^{2} from both sides.

$$-8x^{2}-4x+28-x^{2}=0$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-4x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-9\times 28=-252$$

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 -252.

$$1,-252 2,-126 3,-84 4,-63 6,-42 7,-36 9,-28 12,-21 14,-18$$

Calculate the sum for each pair.

$$1-252=-251 2-126=-124 3-84=-81 4-63=-59 6-42=-36 7-36=-29 9-28=-19 12-21=-9 14-18=-4$$

The solution is the pair that gives sum -4.

$$a=14 b=-18$$

Rewrite -9x^{2}-4x+28 as \left(-9x^{2}+14x\right)+\left(-18x+28\right).

$$\left(-9x^{2}+14x\right)+\left(-18x+28\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(9x-14\right)-2\left(9x-14\right)$$

Factor out common term 9x-14 by using distributive property.

$$\left(9x-14\right)\left(-x-2\right)$$

To find equation solutions, solve 9x-14=0 and -x-2=0.

$$x=\frac{14}{9} x=-2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+2 by -8x+12 and combine like terms.

$$-8x^{2}-4x+24+4=x^{2}$$

Add 24 and 4 to get 28.

$$-8x^{2}-4x+28=x^{2}$$

Subtract x^{2} from both sides.

$$-8x^{2}-4x+28-x^{2}=0$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-4x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -4 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-9\right)\times 28}}{2\left(-9\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-9\right)\times 28}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-4\right)±\sqrt{16+36\times 28}}{2\left(-9\right)}$$

Multiply 36 times 28.

$$x=\frac{-\left(-4\right)±\sqrt{16+1008}}{2\left(-9\right)}$$

Add 16 to 1008.

$$x=\frac{-\left(-4\right)±\sqrt{1024}}{2\left(-9\right)}$$

Take the square root of 1024.

$$x=\frac{-\left(-4\right)±32}{2\left(-9\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±32}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{4±32}{-18}$$

Now solve the equation x=\frac{4±32}{-18} when ± is plus. Add 4 to 32.

$$x=\frac{36}{-18}$$

Divide 36 by -18.

$$x=-2$$

Now solve the equation x=\frac{4±32}{-18} when ± is minus. Subtract 32 from 4.

$$x=-\frac{28}{-18}$$

Reduce the fraction \frac{-28}{-18} to lowest terms by extracting and canceling out 2.

$$x=\frac{14}{9}$$

The equation is now solved.

$$x=-2 x=\frac{14}{9}$$

Steps for Completing the Square

Use the distributive property to multiply x+2 by -8x+12 and combine like terms.

$$-8x^{2}-4x+24+4=x^{2}$$

Add 24 and 4 to get 28.

$$-8x^{2}-4x+28=x^{2}$$

Subtract x^{2} from both sides.

$$-8x^{2}-4x+28-x^{2}=0$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$-9x^{2}-4x+28=0$$

Subtract 28 from both sides. Anything subtracted from zero gives its negation.

$$-9x^{2}-4x=-28$$

Divide both sides by -9.

$$\frac{-9x^{2}-4x}{-9}=-\frac{28}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{4}{-9}\right)x=-\frac{28}{-9}$$

Divide -4 by -9.

$$x^{2}+\frac{4}{9}x=-\frac{28}{-9}$$

Divide -28 by -9.

$$x^{2}+\frac{4}{9}x=\frac{28}{9}$$

Divide \frac{4}{9}, the coefficient of the x term, by 2 to get \frac{2}{9}. Then add the square of \frac{2}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{9}x+\left(\frac{2}{9}\right)^{2}=\frac{28}{9}+\left(\frac{2}{9}\right)^{2}$$

Square \frac{2}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{9}x+\frac{4}{81}=\frac{28}{9}+\frac{4}{81}$$

Add \frac{28}{9} to \frac{4}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{9}x+\frac{4}{81}=\frac{256}{81}$$

Factor x^{2}+\frac{4}{9}x+\frac{4}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{9}\right)^{2}=\frac{256}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{9}\right)^{2}}=\sqrt{\frac{256}{81}}$$

Simplify.

$$x+\frac{2}{9}=\frac{16}{9} x+\frac{2}{9}=-\frac{16}{9}$$

Subtract \frac{2}{9} from both sides of the equation.

$$x=\frac{14}{9} x=-2$$