Solve (3x+1)(1-6x)-(3x+7)(3x+1)=0

$$( 3 x + 1 ) ( 1 - 6 x ) - ( 3 x + 7 ) ( 3 x + 1 ) = 0$$

Solve for x

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

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

Steps Using Factoring By Grouping

Use the distributive property to multiply 3x+1 by 1-6x and combine like terms.

$$-3x-18x^{2}+1-\left(3x+7\right)\left(3x+1\right)=0$$

Use the distributive property to multiply 3x+7 by 3x+1 and combine like terms.

$$-3x-18x^{2}+1-\left(9x^{2}+24x+7\right)=0$$

To find the opposite of 9x^{2}+24x+7, find the opposite of each term.

$$-3x-18x^{2}+1-9x^{2}-24x-7=0$$

Combine -18x^{2} and -9x^{2} to get -27x^{2}.

$$-3x-27x^{2}+1-24x-7=0$$

Combine -3x and -24x to get -27x.

$$-27x-27x^{2}+1-7=0$$

Subtract 7 from 1 to get -6.

$$-27x-27x^{2}-6=0$$

Divide both sides by 3.

$$-9x-9x^{2}-2=0$$

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

$$-9x^{2}-9x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-9\left(-2\right)=18$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -9.

$$a=-3 b=-6$$

Rewrite -9x^{2}-9x-2 as \left(-9x^{2}-3x\right)+\left(-6x-2\right).

$$\left(-9x^{2}-3x\right)+\left(-6x-2\right)$$

Factor out 3x in the first and 2 in the second group.

$$3x\left(-3x-1\right)+2\left(-3x-1\right)$$

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

$$\left(-3x-1\right)\left(3x+2\right)$$

To find equation solutions, solve -3x-1=0 and 3x+2=0.

$$x=-\frac{1}{3} x=-\frac{2}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x+1 by 1-6x and combine like terms.

$$-3x-18x^{2}+1-\left(3x+7\right)\left(3x+1\right)=0$$

Use the distributive property to multiply 3x+7 by 3x+1 and combine like terms.

$$-3x-18x^{2}+1-\left(9x^{2}+24x+7\right)=0$$

To find the opposite of 9x^{2}+24x+7, find the opposite of each term.

$$-3x-18x^{2}+1-9x^{2}-24x-7=0$$

Combine -18x^{2} and -9x^{2} to get -27x^{2}.

$$-3x-27x^{2}+1-24x-7=0$$

Combine -3x and -24x to get -27x.

$$-27x-27x^{2}+1-7=0$$

Subtract 7 from 1 to get -6.

$$-27x-27x^{2}-6=0$$

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.

$$-27x^{2}-27x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -27 for a, -27 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-27\right)\left(-6\right)}}{2\left(-27\right)}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-27\right)\left(-6\right)}}{2\left(-27\right)}$$

Multiply -4 times -27.

$$x=\frac{-\left(-27\right)±\sqrt{729+108\left(-6\right)}}{2\left(-27\right)}$$

Multiply 108 times -6.

$$x=\frac{-\left(-27\right)±\sqrt{729-648}}{2\left(-27\right)}$$

Add 729 to -648.

$$x=\frac{-\left(-27\right)±\sqrt{81}}{2\left(-27\right)}$$

Take the square root of 81.

$$x=\frac{-\left(-27\right)±9}{2\left(-27\right)}$$

The opposite of -27 is 27.

$$x=\frac{27±9}{2\left(-27\right)}$$

Multiply 2 times -27.

$$x=\frac{27±9}{-54}$$

Now solve the equation x=\frac{27±9}{-54} when ± is plus. Add 27 to 9.

$$x=\frac{36}{-54}$$

Reduce the fraction \frac{36}{-54} to lowest terms by extracting and canceling out 18.

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

Now solve the equation x=\frac{27±9}{-54} when ± is minus. Subtract 9 from 27.

$$x=\frac{18}{-54}$$

Reduce the fraction \frac{18}{-54} to lowest terms by extracting and canceling out 18.

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

The equation is now solved.

$$x=-\frac{2}{3} x=-\frac{1}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 3x+1 by 1-6x and combine like terms.

$$-3x-18x^{2}+1-\left(3x+7\right)\left(3x+1\right)=0$$

Use the distributive property to multiply 3x+7 by 3x+1 and combine like terms.

$$-3x-18x^{2}+1-\left(9x^{2}+24x+7\right)=0$$

To find the opposite of 9x^{2}+24x+7, find the opposite of each term.

$$-3x-18x^{2}+1-9x^{2}-24x-7=0$$

Combine -18x^{2} and -9x^{2} to get -27x^{2}.

$$-3x-27x^{2}+1-24x-7=0$$

Combine -3x and -24x to get -27x.

$$-27x-27x^{2}+1-7=0$$

Subtract 7 from 1 to get -6.

$$-27x-27x^{2}-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$-27x-27x^{2}=6$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-27x^{2}-27x=6$$

Divide both sides by -27.

$$\frac{-27x^{2}-27x}{-27}=\frac{6}{-27}$$

Dividing by -27 undoes the multiplication by -27.

$$x^{2}+\left(-\frac{27}{-27}\right)x=\frac{6}{-27}$$

Divide -27 by -27.

$$x^{2}+x=\frac{6}{-27}$$

Reduce the fraction \frac{6}{-27} to lowest terms by extracting and canceling out 3.

$$x^{2}+x=-\frac{2}{9}$$

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=-\frac{2}{9}+\frac{1}{4}$$

Add -\frac{2}{9} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{1}{36}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$x+\frac{1}{2}=\frac{1}{6} x+\frac{1}{2}=-\frac{1}{6}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=-\frac{1}{3} x=-\frac{2}{3}$$