Solve 6x^2+28x-2=2x-10

$$6 x ^ { 2 } + 28 x - 2 = 2 x - 10$$

Solve for x

$$x=-4$$

$$x=-\frac{1}{3}\approx -0.333333333$$

Steps Using Factoring By Grouping

Subtract 2x from both sides.

$$6x^{2}+28x-2-2x=-10$$

Combine 28x and -2x to get 26x.

$$6x^{2}+26x-2=-10$$

Add 10 to both sides.

$$6x^{2}+26x-2+10=0$$

Add -2 and 10 to get 8.

$$6x^{2}+26x+8=0$$

Divide both sides by 2.

$$3x^{2}+13x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=13 ab=3\times 4=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 13.

$$a=1 b=12$$

Rewrite 3x^{2}+13x+4 as \left(3x^{2}+x\right)+\left(12x+4\right).

$$\left(3x^{2}+x\right)+\left(12x+4\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(3x+1\right)+4\left(3x+1\right)$$

Factor out common term 3x+1 by using distributive property.

$$\left(3x+1\right)\left(x+4\right)$$

To find equation solutions, solve 3x+1=0 and x+4=0.

$$x=-\frac{1}{3} x=-4$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$6x^{2}+28x-2-2x=-10$$

Combine 28x and -2x to get 26x.

$$6x^{2}+26x-2=-10$$

Add 10 to both sides.

$$6x^{2}+26x-2+10=0$$

Add -2 and 10 to get 8.

$$6x^{2}+26x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 26 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26±\sqrt{26^{2}-4\times 6\times 8}}{2\times 6}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\times 6\times 8}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-26±\sqrt{676-24\times 8}}{2\times 6}$$

Multiply -24 times 8.

$$x=\frac{-26±\sqrt{676-192}}{2\times 6}$$

Add 676 to -192.

$$x=\frac{-26±\sqrt{484}}{2\times 6}$$

Take the square root of 484.

$$x=\frac{-26±22}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-26±22}{12}$$

Now solve the equation x=\frac{-26±22}{12} when ± is plus. Add -26 to 22.

$$x=-\frac{4}{12}$$

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

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

Now solve the equation x=\frac{-26±22}{12} when ± is minus. Subtract 22 from -26.

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

Divide -48 by 12.

$$x=-4$$

The equation is now solved.

$$x=-\frac{1}{3} x=-4$$

Steps for Completing the Square

Subtract 2x from both sides.

$$6x^{2}+28x-2-2x=-10$$

Combine 28x and -2x to get 26x.

$$6x^{2}+26x-2=-10$$

Add 2 to both sides.

$$6x^{2}+26x=-10+2$$

Add -10 and 2 to get -8.

$$6x^{2}+26x=-8$$

Divide both sides by 6.

$$\frac{6x^{2}+26x}{6}=-\frac{8}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{26}{6}x=-\frac{8}{6}$$

Reduce the fraction \frac{26}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{13}{3}x=-\frac{8}{6}$$

Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{13}{3}x=-\frac{4}{3}$$

Divide \frac{13}{3}, the coefficient of the x term, by 2 to get \frac{13}{6}. Then add the square of \frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{3}x+\left(\frac{13}{6}\right)^{2}=-\frac{4}{3}+\left(\frac{13}{6}\right)^{2}$$

Square \frac{13}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{3}x+\frac{169}{36}=-\frac{4}{3}+\frac{169}{36}$$

Add -\frac{4}{3} to \frac{169}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{3}x+\frac{169}{36}=\frac{121}{36}$$

Factor x^{2}+\frac{13}{3}x+\frac{169}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{6}\right)^{2}=\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Simplify.

$$x+\frac{13}{6}=\frac{11}{6} x+\frac{13}{6}=-\frac{11}{6}$$

Subtract \frac{13}{6} from both sides of the equation.

$$x=-\frac{1}{3} x=-4$$