Solve -11x=x^2+28

$$- 11 x = x ^ { 2 } + 28$$

Solve for x

$$x=-7$$

$$x=-4$$

Steps Using Factoring By Grouping

Subtract x^{2} from both sides.

$$-11x-x^{2}=28$$

Subtract 28 from both sides.

$$-11x-x^{2}-28=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-11x-28=0$$

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

$$a+b=-11 ab=-\left(-28\right)=28$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 28.

$$-1,-28 -2,-14 -4,-7$$

Calculate the sum for each pair.

$$-1-28=-29 -2-14=-16 -4-7=-11$$

The solution is the pair that gives sum -11.

$$a=-4 b=-7$$

Rewrite -x^{2}-11x-28 as \left(-x^{2}-4x\right)+\left(-7x-28\right).

$$\left(-x^{2}-4x\right)+\left(-7x-28\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x-4\right)+7\left(-x-4\right)$$

Factor out common term -x-4 by using distributive property.

$$\left(-x-4\right)\left(x+7\right)$$

To find equation solutions, solve -x-4=0 and x+7=0.

$$x=-4 x=-7$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$-11x-x^{2}=28$$

Subtract 28 from both sides.

$$-11x-x^{2}-28=0$$

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.

$$-x^{2}-11x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -11 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$x=\frac{-\left(-11\right)±\sqrt{121-112}}{2\left(-1\right)}$$

Add 121 to -112.

$$x=\frac{-\left(-11\right)±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

$$x=\frac{-\left(-11\right)±3}{2\left(-1\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{11±3}{-2}$$

Now solve the equation x=\frac{11±3}{-2} when ± is plus. Add 11 to 3.

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

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{11±3}{-2} when ± is minus. Subtract 3 from 11.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

The equation is now solved.

$$x=-7 x=-4$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$-11x-x^{2}=28$$

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.

$$-x^{2}-11x=28$$

Divide both sides by -1.

$$\frac{-x^{2}-11x}{-1}=\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{11}{-1}\right)x=\frac{28}{-1}$$

Divide -11 by -1.

$$x^{2}+11x=\frac{28}{-1}$$

Divide 28 by -1.

$$x^{2}+11x=-28$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-28+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-28+\frac{121}{4}$$

Add -28 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{9}{4}$$

Factor x^{2}+11x+\frac{121}{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}.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{3}{2} x+\frac{11}{2}=-\frac{3}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-4 x=-7$$