Solve -3x-6x^2=-108

$$-3x-6 { x }^{ 2 } = -108$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=4$$

Steps Using Factoring By Grouping

Add 108 to both sides.

$$-3x-6x^{2}+108=0$$

Divide both sides by 3.

$$-x-2x^{2}+36=0$$

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

$$-2x^{2}-x+36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+36. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-2\times 36=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -1.

$$a=8 b=-9$$

Rewrite -2x^{2}-x+36 as \left(-2x^{2}+8x\right)+\left(-9x+36\right).

$$\left(-2x^{2}+8x\right)+\left(-9x+36\right)$$

Factor out 2x in the first and 9 in the second group.

$$2x\left(-x+4\right)+9\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(2x+9\right)$$

To find equation solutions, solve -x+4=0 and 2x+9=0.

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

Steps Using the Quadratic Formula

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.

$$-6x^{2}-3x=-108$$

Add 108 to both sides of the equation.

$$-6x^{2}-3x-\left(-108\right)=-108-\left(-108\right)$$

Subtracting -108 from itself leaves 0.

$$-6x^{2}-3x-\left(-108\right)=0$$

Subtract -108 from 0.

$$-6x^{2}-3x+108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -3 for b, and 108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-6\right)\times 108}}{2\left(-6\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-6\right)\times 108}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-3\right)±\sqrt{9+24\times 108}}{2\left(-6\right)}$$

Multiply 24 times 108.

$$x=\frac{-\left(-3\right)±\sqrt{9+2592}}{2\left(-6\right)}$$

Add 9 to 2592.

$$x=\frac{-\left(-3\right)±\sqrt{2601}}{2\left(-6\right)}$$

Take the square root of 2601.

$$x=\frac{-\left(-3\right)±51}{2\left(-6\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±51}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{3±51}{-12}$$

Now solve the equation x=\frac{3±51}{-12} when ± is plus. Add 3 to 51.

$$x=\frac{54}{-12}$$

Reduce the fraction \frac{54}{-12} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{3±51}{-12} when ± is minus. Subtract 51 from 3.

$$x=-\frac{48}{-12}$$

Divide -48 by -12.

$$x=4$$

The equation is now solved.

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

Steps for Completing the Square

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.

$$-6x^{2}-3x=-108$$

Divide both sides by -6.

$$\frac{-6x^{2}-3x}{-6}=-\frac{108}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\left(-\frac{3}{-6}\right)x=-\frac{108}{-6}$$

Reduce the fraction \frac{-3}{-6} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{2}x=-\frac{108}{-6}$$

Divide -108 by -6.

$$x^{2}+\frac{1}{2}x=18$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=18+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=18+\frac{1}{16}$$

Add 18 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{289}{16}$$

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

$$\left(x+\frac{1}{4}\right)^{2}=\frac{289}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{289}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{17}{4} x+\frac{1}{4}=-\frac{17}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

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