Solve -(3x+1)(5x-6)=0

$$- ( 3 x + 1 ) ( 5 x - 6 ) = 0$$

Solve for x

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

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

Steps Using Factoring By Grouping

To find the opposite of 3x+1, find the opposite of each term.

$$\left(-3x-1\right)\left(5x-6\right)=0$$

Apply the distributive property by multiplying each term of -3x-1 by each term of 5x-6.

$$-15x^{2}+18x-5x+6=0$$

Combine 18x and -5x to get 13x.

$$-15x^{2}+13x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -15x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=13 ab=-15\times 6=-90$$

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 -90.

$$-1,90 -2,45 -3,30 -5,18 -6,15 -9,10$$

Calculate the sum for each pair.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

The solution is the pair that gives sum 13.

$$a=18 b=-5$$

Rewrite -15x^{2}+13x+6 as \left(-15x^{2}+18x\right)+\left(-5x+6\right).

$$\left(-15x^{2}+18x\right)+\left(-5x+6\right)$$

Factor out 3x in -15x^{2}+18x.

$$3x\left(-5x+6\right)-5x+6$$

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

$$\left(-5x+6\right)\left(3x+1\right)$$

To find equation solutions, solve -5x+6=0 and 3x+1=0.

$$x=\frac{6}{5} x=-\frac{1}{3}$$

Steps Using the Quadratic Formula

To find the opposite of 3x+1, find the opposite of each term.

$$\left(-3x-1\right)\left(5x-6\right)=0$$

Apply the distributive property by multiplying each term of -3x-1 by each term of 5x-6.

$$-15x^{2}+18x-5x+6=0$$

Combine 18x and -5x to get 13x.

$$-15x^{2}+13x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 13 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-15\right)\times 6}}{2\left(-15\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-15\right)\times 6}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-13±\sqrt{169+60\times 6}}{2\left(-15\right)}$$

Multiply 60 times 6.

$$x=\frac{-13±\sqrt{169+360}}{2\left(-15\right)}$$

Add 169 to 360.

$$x=\frac{-13±\sqrt{529}}{2\left(-15\right)}$$

Take the square root of 529.

$$x=\frac{-13±23}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-13±23}{-30}$$

Now solve the equation x=\frac{-13±23}{-30} when ± is plus. Add -13 to 23.

$$x=\frac{10}{-30}$$

Reduce the fraction \frac{10}{-30} to lowest terms by extracting and canceling out 10.

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

Now solve the equation x=\frac{-13±23}{-30} when ± is minus. Subtract 23 from -13.

$$x=-\frac{36}{-30}$$

Reduce the fraction \frac{-36}{-30} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=-\frac{1}{3} x=\frac{6}{5}$$

Steps for Completing the Square

To find the opposite of 3x+1, find the opposite of each term.

$$\left(-3x-1\right)\left(5x-6\right)=0$$

Apply the distributive property by multiplying each term of -3x-1 by each term of 5x-6.

$$-15x^{2}+18x-5x+6=0$$

Combine 18x and -5x to get 13x.

$$-15x^{2}+13x+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-15x^{2}+13x=-6$$

Divide both sides by -15.

$$\frac{-15x^{2}+13x}{-15}=-\frac{6}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{13}{-15}x=-\frac{6}{-15}$$

Divide 13 by -15.

$$x^{2}-\frac{13}{15}x=-\frac{6}{-15}$$

Reduce the fraction \frac{-6}{-15} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{13}{15}x=\frac{2}{5}$$

Divide -\frac{13}{15}, the coefficient of the x term, by 2 to get -\frac{13}{30}. Then add the square of -\frac{13}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{15}x+\left(-\frac{13}{30}\right)^{2}=\frac{2}{5}+\left(-\frac{13}{30}\right)^{2}$$

Square -\frac{13}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{15}x+\frac{169}{900}=\frac{2}{5}+\frac{169}{900}$$

Add \frac{2}{5} to \frac{169}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{15}x+\frac{169}{900}=\frac{529}{900}$$

Factor x^{2}-\frac{13}{15}x+\frac{169}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{30}\right)^{2}=\frac{529}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{30}\right)^{2}}=\sqrt{\frac{529}{900}}$$

Simplify.

$$x-\frac{13}{30}=\frac{23}{30} x-\frac{13}{30}=-\frac{23}{30}$$

Add \frac{13}{30} to both sides of the equation.

$$x=\frac{6}{5} x=-\frac{1}{3}$$