Solve 0=6x^2+11x-10

$$0 = 6 { x }^{ 2 } +11x-10$$

Solve for x

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

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$6x^{2}+11x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=11 ab=6\left(-10\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 11.

$$a=-4 b=15$$

Rewrite 6x^{2}+11x-10 as \left(6x^{2}-4x\right)+\left(15x-10\right).

$$\left(6x^{2}-4x\right)+\left(15x-10\right)$$

Factor out 2x in the first and 5 in the second group.

$$2x\left(3x-2\right)+5\left(3x-2\right)$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(2x+5\right)$$

To find equation solutions, solve 3x-2=0 and 2x+5=0.

$$x=\frac{2}{3} x=-\frac{5}{2}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$6x^{2}+11x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 11 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 6\left(-10\right)}}{2\times 6}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 6\left(-10\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-11±\sqrt{121-24\left(-10\right)}}{2\times 6}$$

Multiply -24 times -10.

$$x=\frac{-11±\sqrt{121+240}}{2\times 6}$$

Add 121 to 240.

$$x=\frac{-11±\sqrt{361}}{2\times 6}$$

Take the square root of 361.

$$x=\frac{-11±19}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-11±19}{12}$$

Now solve the equation x=\frac{-11±19}{12} when ± is plus. Add -11 to 19.

$$x=\frac{8}{12}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{-11±19}{12} when ± is minus. Subtract 19 from -11.

$$x=-\frac{30}{12}$$

Reduce the fraction \frac{-30}{12} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=\frac{2}{3} x=-\frac{5}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$6x^{2}+11x-10=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$6x^{2}+11x=10$$

Divide both sides by 6.

$$\frac{6x^{2}+11x}{6}=\frac{10}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{11}{6}x=\frac{10}{6}$$

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{11}{6}x=\frac{5}{3}$$

Divide \frac{11}{6}, the coefficient of the x term, by 2 to get \frac{11}{12}. Then add the square of \frac{11}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{6}x+\left(\frac{11}{12}\right)^{2}=\frac{5}{3}+\left(\frac{11}{12}\right)^{2}$$

Square \frac{11}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{5}{3}+\frac{121}{144}$$

Add \frac{5}{3} to \frac{121}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{361}{144}$$

Factor x^{2}+\frac{11}{6}x+\frac{121}{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}.

$$\left(x+\frac{11}{12}\right)^{2}=\frac{361}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{12}\right)^{2}}=\sqrt{\frac{361}{144}}$$

Simplify.

$$x+\frac{11}{12}=\frac{19}{12} x+\frac{11}{12}=-\frac{19}{12}$$

Subtract \frac{11}{12} from both sides of the equation.

$$x=\frac{2}{3} x=-\frac{5}{2}$$