Solve 2(9x^2+6x+1)=(3x+1)(x-2)

$$2 ( 9 x ^ { 2 } + 6 x + 1 ) = ( 3 x + 1 ) ( x - 2 )$$

Solve for x

$$x=-\frac{4}{5}=-0.8$$

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

Steps Using Factoring By Grouping

Use the distributive property to multiply 2 by 9x^{2}+6x+1.

$$18x^{2}+12x+2=\left(3x+1\right)\left(x-2\right)$$

Use the distributive property to multiply 3x+1 by x-2 and combine like terms.

$$18x^{2}+12x+2=3x^{2}-5x-2$$

Subtract 3x^{2} from both sides.

$$18x^{2}+12x+2-3x^{2}=-5x-2$$

Combine 18x^{2} and -3x^{2} to get 15x^{2}.

$$15x^{2}+12x+2=-5x-2$$

Add 5x to both sides.

$$15x^{2}+12x+2+5x=-2$$

Combine 12x and 5x to get 17x.

$$15x^{2}+17x+2=-2$$

Add 2 to both sides.

$$15x^{2}+17x+2+2=0$$

Add 2 and 2 to get 4.

$$15x^{2}+17x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 15x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=17 ab=15\times 4=60$$

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 60.

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

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 17.

$$a=5 b=12$$

Rewrite 15x^{2}+17x+4 as \left(15x^{2}+5x\right)+\left(12x+4\right).

$$\left(15x^{2}+5x\right)+\left(12x+4\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(3x+1\right)+4\left(3x+1\right)$$

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

$$\left(3x+1\right)\left(5x+4\right)$$

To find equation solutions, solve 3x+1=0 and 5x+4=0.

$$x=-\frac{1}{3} x=-\frac{4}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by 9x^{2}+6x+1.

$$18x^{2}+12x+2=\left(3x+1\right)\left(x-2\right)$$

Use the distributive property to multiply 3x+1 by x-2 and combine like terms.

$$18x^{2}+12x+2=3x^{2}-5x-2$$

Subtract 3x^{2} from both sides.

$$18x^{2}+12x+2-3x^{2}=-5x-2$$

Combine 18x^{2} and -3x^{2} to get 15x^{2}.

$$15x^{2}+12x+2=-5x-2$$

Add 5x to both sides.

$$15x^{2}+12x+2+5x=-2$$

Combine 12x and 5x to get 17x.

$$15x^{2}+17x+2=-2$$

Add 2 to both sides.

$$15x^{2}+17x+2+2=0$$

Add 2 and 2 to get 4.

$$15x^{2}+17x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 17 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 15\times 4}}{2\times 15}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 15\times 4}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-17±\sqrt{289-60\times 4}}{2\times 15}$$

Multiply -60 times 4.

$$x=\frac{-17±\sqrt{289-240}}{2\times 15}$$

Add 289 to -240.

$$x=\frac{-17±\sqrt{49}}{2\times 15}$$

Take the square root of 49.

$$x=\frac{-17±7}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-17±7}{30}$$

Now solve the equation x=\frac{-17±7}{30} when ± is plus. Add -17 to 7.

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

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{-17±7}{30} when ± is minus. Subtract 7 from -17.

$$x=-\frac{24}{30}$$

Reduce the fraction \frac{-24}{30} to lowest terms by extracting and canceling out 6.

$$x=-\frac{4}{5}$$

The equation is now solved.

$$x=-\frac{1}{3} x=-\frac{4}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 2 by 9x^{2}+6x+1.

$$18x^{2}+12x+2=\left(3x+1\right)\left(x-2\right)$$

Use the distributive property to multiply 3x+1 by x-2 and combine like terms.

$$18x^{2}+12x+2=3x^{2}-5x-2$$

Subtract 3x^{2} from both sides.

$$18x^{2}+12x+2-3x^{2}=-5x-2$$

Combine 18x^{2} and -3x^{2} to get 15x^{2}.

$$15x^{2}+12x+2=-5x-2$$

Add 5x to both sides.

$$15x^{2}+12x+2+5x=-2$$

Combine 12x and 5x to get 17x.

$$15x^{2}+17x+2=-2$$

Subtract 2 from both sides.

$$15x^{2}+17x=-2-2$$

Subtract 2 from -2 to get -4.

$$15x^{2}+17x=-4$$

Divide both sides by 15.

$$\frac{15x^{2}+17x}{15}=-\frac{4}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\frac{17}{15}x=-\frac{4}{15}$$

Divide \frac{17}{15}, the coefficient of the x term, by 2 to get \frac{17}{30}. Then add the square of \frac{17}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{17}{15}x+\left(\frac{17}{30}\right)^{2}=-\frac{4}{15}+\left(\frac{17}{30}\right)^{2}$$

Square \frac{17}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{17}{15}x+\frac{289}{900}=-\frac{4}{15}+\frac{289}{900}$$

Add -\frac{4}{15} to \frac{289}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{17}{15}x+\frac{289}{900}=\frac{49}{900}$$

Factor x^{2}+\frac{17}{15}x+\frac{289}{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}.

$$\left(x+\frac{17}{30}\right)^{2}=\frac{49}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{30}\right)^{2}}=\sqrt{\frac{49}{900}}$$

Simplify.

$$x+\frac{17}{30}=\frac{7}{30} x+\frac{17}{30}=-\frac{7}{30}$$

Subtract \frac{17}{30} from both sides of the equation.

$$x=-\frac{1}{3} x=-\frac{4}{5}$$