Solve 66u=12-78u^2

$$66 u = 12 - 78 u ^ { 2 }$$

Solve for u

$$u=-1$$

$$u=\frac{2}{13}\approx 0.153846154$$

Steps Using Factoring By Grouping

Subtract 12 from both sides.

$$66u-12=-78u^{2}$$

Add 78u^{2} to both sides.

$$66u-12+78u^{2}=0$$

Divide both sides by 6.

$$11u-2+13u^{2}=0$$

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

$$13u^{2}+11u-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 13u^{2}+au+bu-2. To find a and b, set up a system to be solved.

$$a+b=11 ab=13\left(-2\right)=-26$$

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 -26.

$$-1,26 -2,13$$

Calculate the sum for each pair.

$$-1+26=25 -2+13=11$$

The solution is the pair that gives sum 11.

$$a=-2 b=13$$

Rewrite 13u^{2}+11u-2 as \left(13u^{2}-2u\right)+\left(13u-2\right).

$$\left(13u^{2}-2u\right)+\left(13u-2\right)$$

Factor out u in 13u^{2}-2u.

$$u\left(13u-2\right)+13u-2$$

Factor out common term 13u-2 by using distributive property.

$$\left(13u-2\right)\left(u+1\right)$$

To find equation solutions, solve 13u-2=0 and u+1=0.

$$u=\frac{2}{13} u=-1$$

Steps Using the Quadratic Formula

Subtract 12 from both sides.

$$66u-12=-78u^{2}$$

Add 78u^{2} to both sides.

$$66u-12+78u^{2}=0$$

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.

$$78u^{2}+66u-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 78 for a, 66 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-66±\sqrt{66^{2}-4\times 78\left(-12\right)}}{2\times 78}$$

Square 66.

$$u=\frac{-66±\sqrt{4356-4\times 78\left(-12\right)}}{2\times 78}$$

Multiply -4 times 78.

$$u=\frac{-66±\sqrt{4356-312\left(-12\right)}}{2\times 78}$$

Multiply -312 times -12.

$$u=\frac{-66±\sqrt{4356+3744}}{2\times 78}$$

Add 4356 to 3744.

$$u=\frac{-66±\sqrt{8100}}{2\times 78}$$

Take the square root of 8100.

$$u=\frac{-66±90}{2\times 78}$$

Multiply 2 times 78.

$$u=\frac{-66±90}{156}$$

Now solve the equation u=\frac{-66±90}{156} when ± is plus. Add -66 to 90.

$$u=\frac{24}{156}$$

Reduce the fraction \frac{24}{156} to lowest terms by extracting and canceling out 12.

$$u=\frac{2}{13}$$

Now solve the equation u=\frac{-66±90}{156} when ± is minus. Subtract 90 from -66.

$$u=-\frac{156}{156}$$

Divide -156 by 156.

$$u=-1$$

The equation is now solved.

$$u=\frac{2}{13} u=-1$$

Steps for Completing the Square

Add 78u^{2} to both sides.

$$66u+78u^{2}=12$$

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.

$$78u^{2}+66u=12$$

Divide both sides by 78.

$$\frac{78u^{2}+66u}{78}=\frac{12}{78}$$

Dividing by 78 undoes the multiplication by 78.

$$u^{2}+\frac{66}{78}u=\frac{12}{78}$$

Reduce the fraction \frac{66}{78} to lowest terms by extracting and canceling out 6.

$$u^{2}+\frac{11}{13}u=\frac{12}{78}$$

Reduce the fraction \frac{12}{78} to lowest terms by extracting and canceling out 6.

$$u^{2}+\frac{11}{13}u=\frac{2}{13}$$

Divide \frac{11}{13}, the coefficient of the x term, by 2 to get \frac{11}{26}. Then add the square of \frac{11}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}+\frac{11}{13}u+\left(\frac{11}{26}\right)^{2}=\frac{2}{13}+\left(\frac{11}{26}\right)^{2}$$

Square \frac{11}{26} by squaring both the numerator and the denominator of the fraction.

$$u^{2}+\frac{11}{13}u+\frac{121}{676}=\frac{2}{13}+\frac{121}{676}$$

Add \frac{2}{13} to \frac{121}{676} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$u^{2}+\frac{11}{13}u+\frac{121}{676}=\frac{225}{676}$$

Factor u^{2}+\frac{11}{13}u+\frac{121}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u+\frac{11}{26}\right)^{2}=\frac{225}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u+\frac{11}{26}\right)^{2}}=\sqrt{\frac{225}{676}}$$

Simplify.

$$u+\frac{11}{26}=\frac{15}{26} u+\frac{11}{26}=-\frac{15}{26}$$

Subtract \frac{11}{26} from both sides of the equation.

$$u=\frac{2}{13} u=-1$$