Solve 12x^2=240-66x

$$12 x ^ { 2 } = 240 - 66 x$$

Solve for x

$$x=-8$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Factoring By Grouping

Subtract 240 from both sides.

$$12x^{2}-240=-66x$$

Add 66x to both sides.

$$12x^{2}-240+66x=0$$

Divide both sides by 6.

$$2x^{2}-40+11x=0$$

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

$$2x^{2}+11x-40=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-40. To find a and b, set up a system to be solved.

$$a+b=11 ab=2\left(-40\right)=-80$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -80.

$$-1,80 -2,40 -4,20 -5,16 -8,10$$

Calculate the sum for each pair.

$$-1+80=79 -2+40=38 -4+20=16 -5+16=11 -8+10=2$$

The solution is the pair that gives sum 11.

$$a=-5 b=16$$

Rewrite 2x^{2}+11x-40 as \left(2x^{2}-5x\right)+\left(16x-40\right).

$$\left(2x^{2}-5x\right)+\left(16x-40\right)$$

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

$$x\left(2x-5\right)+8\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(x+8\right)$$

To find equation solutions, solve 2x-5=0 and x+8=0.

$$x=\frac{5}{2} x=-8$$

Steps Using the Quadratic Formula

Subtract 240 from both sides.

$$12x^{2}-240=-66x$$

Add 66x to both sides.

$$12x^{2}-240+66x=0$$

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.

$$12x^{2}+66x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 66 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-66±\sqrt{66^{2}-4\times 12\left(-240\right)}}{2\times 12}$$

Square 66.

$$x=\frac{-66±\sqrt{4356-4\times 12\left(-240\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-66±\sqrt{4356-48\left(-240\right)}}{2\times 12}$$

Multiply -48 times -240.

$$x=\frac{-66±\sqrt{4356+11520}}{2\times 12}$$

Add 4356 to 11520.

$$x=\frac{-66±\sqrt{15876}}{2\times 12}$$

Take the square root of 15876.

$$x=\frac{-66±126}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-66±126}{24}$$

Now solve the equation x=\frac{-66±126}{24} when ± is plus. Add -66 to 126.

$$x=\frac{60}{24}$$

Reduce the fraction \frac{60}{24} to lowest terms by extracting and canceling out 12.

$$x=\frac{5}{2}$$

Now solve the equation x=\frac{-66±126}{24} when ± is minus. Subtract 126 from -66.

$$x=-\frac{192}{24}$$

Divide -192 by 24.

$$x=-8$$

The equation is now solved.

$$x=\frac{5}{2} x=-8$$

Steps for Completing the Square

Add 66x to both sides.

$$12x^{2}+66x=240$$

Divide both sides by 12.

$$\frac{12x^{2}+66x}{12}=\frac{240}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{66}{12}x=\frac{240}{12}$$

Reduce the fraction \frac{66}{12} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{11}{2}x=\frac{240}{12}$$

Divide 240 by 12.

$$x^{2}+\frac{11}{2}x=20$$

Divide \frac{11}{2}, the coefficient of the x term, by 2 to get \frac{11}{4}. Then add the square of \frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{2}x+\left(\frac{11}{4}\right)^{2}=20+\left(\frac{11}{4}\right)^{2}$$

Square \frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=20+\frac{121}{16}$$

Add 20 to \frac{121}{16}.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{441}{16}$$

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

$$\left(x+\frac{11}{4}\right)^{2}=\frac{441}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Simplify.

$$x+\frac{11}{4}=\frac{21}{4} x+\frac{11}{4}=-\frac{21}{4}$$

Subtract \frac{11}{4} from both sides of the equation.

$$x=\frac{5}{2} x=-8$$