Solve 0=30x^2+11x-30

$$0 = 30 x ^ { 2 } + 11 x - 30$$

Solve for x

$$x = -\frac{6}{5} = -1\frac{1}{5} = -1.2$$

$$x=\frac{5}{6}\approx 0.833333333$$

Steps Using Factoring By Grouping

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

$$30x^{2}+11x-30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 30x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=11 ab=30\left(-30\right)=-900$$

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 -900.

$$-1,900 -2,450 -3,300 -4,225 -5,180 -6,150 -9,100 -10,90 -12,75 -15,60 -18,50 -20,45 -25,36 -30,30$$

Calculate the sum for each pair.

$$-1+900=899 -2+450=448 -3+300=297 -4+225=221 -5+180=175 -6+150=144 -9+100=91 -10+90=80 -12+75=63 -15+60=45 -18+50=32 -20+45=25 -25+36=11 -30+30=0$$

The solution is the pair that gives sum 11.

$$a=-25 b=36$$

Rewrite 30x^{2}+11x-30 as \left(30x^{2}-25x\right)+\left(36x-30\right).

$$\left(30x^{2}-25x\right)+\left(36x-30\right)$$

Factor out 5x in the first and 6 in the second group.

$$5x\left(6x-5\right)+6\left(6x-5\right)$$

Factor out common term 6x-5 by using distributive property.

$$\left(6x-5\right)\left(5x+6\right)$$

To find equation solutions, solve 6x-5=0 and 5x+6=0.

$$x=\frac{5}{6} x=-\frac{6}{5}$$

Steps Using the Quadratic Formula

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

$$30x^{2}+11x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 30 for a, 11 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 30\left(-30\right)}}{2\times 30}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 30\left(-30\right)}}{2\times 30}$$

Multiply -4 times 30.

$$x=\frac{-11±\sqrt{121-120\left(-30\right)}}{2\times 30}$$

Multiply -120 times -30.

$$x=\frac{-11±\sqrt{121+3600}}{2\times 30}$$

Add 121 to 3600.

$$x=\frac{-11±\sqrt{3721}}{2\times 30}$$

Take the square root of 3721.

$$x=\frac{-11±61}{2\times 30}$$

Multiply 2 times 30.

$$x=\frac{-11±61}{60}$$

Now solve the equation x=\frac{-11±61}{60} when ± is plus. Add -11 to 61.

$$x=\frac{50}{60}$$

Reduce the fraction \frac{50}{60} to lowest terms by extracting and canceling out 10.

$$x=\frac{5}{6}$$

Now solve the equation x=\frac{-11±61}{60} when ± is minus. Subtract 61 from -11.

$$x=-\frac{72}{60}$$

Reduce the fraction \frac{-72}{60} to lowest terms by extracting and canceling out 12.

$$x=-\frac{6}{5}$$

The equation is now solved.

$$x=\frac{5}{6} x=-\frac{6}{5}$$

Steps for Completing the Square

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

$$30x^{2}+11x-30=0$$

Add 30 to both sides. Anything plus zero gives itself.

$$30x^{2}+11x=30$$

Divide both sides by 30.

$$\frac{30x^{2}+11x}{30}=\frac{30}{30}$$

Dividing by 30 undoes the multiplication by 30.

$$x^{2}+\frac{11}{30}x=\frac{30}{30}$$

Divide 30 by 30.

$$x^{2}+\frac{11}{30}x=1$$

Divide \frac{11}{30}, the coefficient of the x term, by 2 to get \frac{11}{60}. Then add the square of \frac{11}{60} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{30}x+\left(\frac{11}{60}\right)^{2}=1+\left(\frac{11}{60}\right)^{2}$$

Square \frac{11}{60} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{30}x+\frac{121}{3600}=1+\frac{121}{3600}$$

Add 1 to \frac{121}{3600}.

$$x^{2}+\frac{11}{30}x+\frac{121}{3600}=\frac{3721}{3600}$$

Factor x^{2}+\frac{11}{30}x+\frac{121}{3600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{60}\right)^{2}=\frac{3721}{3600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{60}\right)^{2}}=\sqrt{\frac{3721}{3600}}$$

Simplify.

$$x+\frac{11}{60}=\frac{61}{60} x+\frac{11}{60}=-\frac{61}{60}$$

Subtract \frac{11}{60} from both sides of the equation.

$$x=\frac{5}{6} x=-\frac{6}{5}$$