Solve 12s^2-16=-94s

$$12 s ^ { 2 } - 16 = - 94 s$$

Solve for s

$$s=-8$$

$$s=\frac{1}{6}\approx 0.166666667$$

Steps Using Factoring By Grouping

Add 94s to both sides.

$$12s^{2}-16+94s=0$$

Divide both sides by 2.

$$6s^{2}-8+47s=0$$

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

$$6s^{2}+47s-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6s^{2}+as+bs-8. To find a and b, set up a system to be solved.

$$a+b=47 ab=6\left(-8\right)=-48$$

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 -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Calculate the sum for each pair.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

The solution is the pair that gives sum 47.

$$a=-1 b=48$$

Rewrite 6s^{2}+47s-8 as \left(6s^{2}-s\right)+\left(48s-8\right).

$$\left(6s^{2}-s\right)+\left(48s-8\right)$$

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

$$s\left(6s-1\right)+8\left(6s-1\right)$$

Factor out common term 6s-1 by using distributive property.

$$\left(6s-1\right)\left(s+8\right)$$

To find equation solutions, solve 6s-1=0 and s+8=0.

$$s=\frac{1}{6} s=-8$$

Steps Using the Quadratic Formula

Add 94s to both sides.

$$12s^{2}-16+94s=0$$

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.

$$12s^{2}+94s-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 94 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-94±\sqrt{94^{2}-4\times 12\left(-16\right)}}{2\times 12}$$

Square 94.

$$s=\frac{-94±\sqrt{8836-4\times 12\left(-16\right)}}{2\times 12}$$

Multiply -4 times 12.

$$s=\frac{-94±\sqrt{8836-48\left(-16\right)}}{2\times 12}$$

Multiply -48 times -16.

$$s=\frac{-94±\sqrt{8836+768}}{2\times 12}$$

Add 8836 to 768.

$$s=\frac{-94±\sqrt{9604}}{2\times 12}$$

Take the square root of 9604.

$$s=\frac{-94±98}{2\times 12}$$

Multiply 2 times 12.

$$s=\frac{-94±98}{24}$$

Now solve the equation s=\frac{-94±98}{24} when ± is plus. Add -94 to 98.

$$s=\frac{4}{24}$$

Reduce the fraction \frac{4}{24} to lowest terms by extracting and canceling out 4.

$$s=\frac{1}{6}$$

Now solve the equation s=\frac{-94±98}{24} when ± is minus. Subtract 98 from -94.

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

Divide -192 by 24.

$$s=-8$$

The equation is now solved.

$$s=\frac{1}{6} s=-8$$

Steps for Completing the Square

Add 94s to both sides.

$$12s^{2}-16+94s=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$12s^{2}+94s=16$$

Divide both sides by 12.

$$\frac{12s^{2}+94s}{12}=\frac{16}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$s^{2}+\frac{94}{12}s=\frac{16}{12}$$

Reduce the fraction \frac{94}{12} to lowest terms by extracting and canceling out 2.

$$s^{2}+\frac{47}{6}s=\frac{16}{12}$$

Reduce the fraction \frac{16}{12} to lowest terms by extracting and canceling out 4.

$$s^{2}+\frac{47}{6}s=\frac{4}{3}$$

Divide \frac{47}{6}, the coefficient of the x term, by 2 to get \frac{47}{12}. Then add the square of \frac{47}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}+\frac{47}{6}s+\left(\frac{47}{12}\right)^{2}=\frac{4}{3}+\left(\frac{47}{12}\right)^{2}$$

Square \frac{47}{12} by squaring both the numerator and the denominator of the fraction.

$$s^{2}+\frac{47}{6}s+\frac{2209}{144}=\frac{4}{3}+\frac{2209}{144}$$

Add \frac{4}{3} to \frac{2209}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$s^{2}+\frac{47}{6}s+\frac{2209}{144}=\frac{2401}{144}$$

Factor s^{2}+\frac{47}{6}s+\frac{2209}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(s+\frac{47}{12}\right)^{2}=\frac{2401}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s+\frac{47}{12}\right)^{2}}=\sqrt{\frac{2401}{144}}$$

Simplify.

$$s+\frac{47}{12}=\frac{49}{12} s+\frac{47}{12}=-\frac{49}{12}$$

Subtract \frac{47}{12} from both sides of the equation.

$$s=\frac{1}{6} s=-8$$