Solve 3x^2-56=-2x

$$3 x ^ { 2 } - 56 = - 2 x$$

Solve for x

$$x = -\frac{14}{3} = -4\frac{2}{3} \approx -4.666666667$$

$$x=4$$

Steps Using Factoring By Grouping

Add 2x to both sides.

$$3x^{2}-56+2x=0$$

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

$$3x^{2}+2x-56=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-56. To find a and b, set up a system to be solved.

$$a+b=2 ab=3\left(-56\right)=-168$$

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 -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Calculate the sum for each pair.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

The solution is the pair that gives sum 2.

$$a=-12 b=14$$

Rewrite 3x^{2}+2x-56 as \left(3x^{2}-12x\right)+\left(14x-56\right).

$$\left(3x^{2}-12x\right)+\left(14x-56\right)$$

Factor out 3x in the first and 14 in the second group.

$$3x\left(x-4\right)+14\left(x-4\right)$$

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

$$\left(x-4\right)\left(3x+14\right)$$

To find equation solutions, solve x-4=0 and 3x+14=0.

$$x=4 x=-\frac{14}{3}$$

Steps Using the Quadratic Formula

Add 2x to both sides.

$$3x^{2}-56+2x=0$$

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.

$$3x^{2}+2x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 2 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\left(-56\right)}}{2\times 3}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 3\left(-56\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-2±\sqrt{4-12\left(-56\right)}}{2\times 3}$$

Multiply -12 times -56.

$$x=\frac{-2±\sqrt{4+672}}{2\times 3}$$

Add 4 to 672.

$$x=\frac{-2±\sqrt{676}}{2\times 3}$$

Take the square root of 676.

$$x=\frac{-2±26}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-2±26}{6}$$

Now solve the equation x=\frac{-2±26}{6} when ± is plus. Add -2 to 26.

$$x=\frac{24}{6}$$

Divide 24 by 6.

$$x=4$$

Now solve the equation x=\frac{-2±26}{6} when ± is minus. Subtract 26 from -2.

$$x=-\frac{28}{6}$$

Reduce the fraction \frac{-28}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{14}{3}$$

The equation is now solved.

$$x=4 x=-\frac{14}{3}$$

Steps for Completing the Square

Add 2x to both sides.

$$3x^{2}-56+2x=0$$

Add 56 to both sides. Anything plus zero gives itself.

$$3x^{2}+2x=56$$

Divide both sides by 3.

$$\frac{3x^{2}+2x}{3}=\frac{56}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{2}{3}x=\frac{56}{3}$$

Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=\frac{56}{3}+\left(\frac{1}{3}\right)^{2}$$

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{56}{3}+\frac{1}{9}$$

Add \frac{56}{3} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{169}{9}$$

Factor x^{2}+\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{169}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{169}{9}}$$

Simplify.

$$x+\frac{1}{3}=\frac{13}{3} x+\frac{1}{3}=-\frac{13}{3}$$

Subtract \frac{1}{3} from both sides of the equation.

$$x=4 x=-\frac{14}{3}$$