Solve -32x=2x^2-x-51

$$- 32 x = 2 x ^ { 2 } - x - 51$$

Solve for x

$$x=-17$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

Subtract 2x^{2} from both sides.

$$-32x-2x^{2}=-x-51$$

Add x to both sides.

$$-32x-2x^{2}+x=-51$$

Combine -32x and x to get -31x.

$$-31x-2x^{2}=-51$$

Add 51 to both sides.

$$-31x-2x^{2}+51=0$$

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

$$-2x^{2}-31x+51=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+51. To find a and b, set up a system to be solved.

$$a+b=-31 ab=-2\times 51=-102$$

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 -102.

$$1,-102 2,-51 3,-34 6,-17$$

Calculate the sum for each pair.

$$1-102=-101 2-51=-49 3-34=-31 6-17=-11$$

The solution is the pair that gives sum -31.

$$a=3 b=-34$$

Rewrite -2x^{2}-31x+51 as \left(-2x^{2}+3x\right)+\left(-34x+51\right).

$$\left(-2x^{2}+3x\right)+\left(-34x+51\right)$$

Factor out -x in the first and -17 in the second group.

$$-x\left(2x-3\right)-17\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(-x-17\right)$$

To find equation solutions, solve 2x-3=0 and -x-17=0.

$$x=\frac{3}{2} x=-17$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$-32x-2x^{2}=-x-51$$

Add x to both sides.

$$-32x-2x^{2}+x=-51$$

Combine -32x and x to get -31x.

$$-31x-2x^{2}=-51$$

Add 51 to both sides.

$$-31x-2x^{2}+51=0$$

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.

$$-2x^{2}-31x+51=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -31 for b, and 51 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\left(-2\right)\times 51}}{2\left(-2\right)}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\left(-2\right)\times 51}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-31\right)±\sqrt{961+8\times 51}}{2\left(-2\right)}$$

Multiply 8 times 51.

$$x=\frac{-\left(-31\right)±\sqrt{961+408}}{2\left(-2\right)}$$

Add 961 to 408.

$$x=\frac{-\left(-31\right)±\sqrt{1369}}{2\left(-2\right)}$$

Take the square root of 1369.

$$x=\frac{-\left(-31\right)±37}{2\left(-2\right)}$$

The opposite of -31 is 31.

$$x=\frac{31±37}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{31±37}{-4}$$

Now solve the equation x=\frac{31±37}{-4} when ± is plus. Add 31 to 37.

$$x=\frac{68}{-4}$$

Divide 68 by -4.

$$x=-17$$

Now solve the equation x=\frac{31±37}{-4} when ± is minus. Subtract 37 from 31.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{2}$$

The equation is now solved.

$$x=-17 x=\frac{3}{2}$$

Steps for Completing the Square

Subtract 2x^{2} from both sides.

$$-32x-2x^{2}=-x-51$$

Add x to both sides.

$$-32x-2x^{2}+x=-51$$

Combine -32x and x to get -31x.

$$-31x-2x^{2}=-51$$

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.

$$-2x^{2}-31x=-51$$

Divide both sides by -2.

$$\frac{-2x^{2}-31x}{-2}=-\frac{51}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{31}{-2}\right)x=-\frac{51}{-2}$$

Divide -31 by -2.

$$x^{2}+\frac{31}{2}x=-\frac{51}{-2}$$

Divide -51 by -2.

$$x^{2}+\frac{31}{2}x=\frac{51}{2}$$

Divide \frac{31}{2}, the coefficient of the x term, by 2 to get \frac{31}{4}. Then add the square of \frac{31}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{31}{2}x+\left(\frac{31}{4}\right)^{2}=\frac{51}{2}+\left(\frac{31}{4}\right)^{2}$$

Square \frac{31}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{31}{2}x+\frac{961}{16}=\frac{51}{2}+\frac{961}{16}$$

Add \frac{51}{2} to \frac{961}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{31}{2}x+\frac{961}{16}=\frac{1369}{16}$$

Factor x^{2}+\frac{31}{2}x+\frac{961}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{31}{4}\right)^{2}=\frac{1369}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{31}{4}\right)^{2}}=\sqrt{\frac{1369}{16}}$$

Simplify.

$$x+\frac{31}{4}=\frac{37}{4} x+\frac{31}{4}=-\frac{37}{4}$$

Subtract \frac{31}{4} from both sides of the equation.

$$x=\frac{3}{2} x=-17$$