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

$$= ( 3 x + 2 ) 2 = ( x - 1 ) ( 9 x + 1 )$$

Solve for x

$$x = \frac{\sqrt{94} + 7}{9} \approx 1.855039968$$

$$x=\frac{7-\sqrt{94}}{9}\approx -0.299484413$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x+2 by 2.

$$6x+4=\left(x-1\right)\left(9x+1\right)$$

Use the distributive property to multiply x-1 by 9x+1 and combine like terms.

$$6x+4=9x^{2}-8x-1$$

Subtract 9x^{2} from both sides.

$$6x+4-9x^{2}=-8x-1$$

Add 8x to both sides.

$$6x+4-9x^{2}+8x=-1$$

Combine 6x and 8x to get 14x.

$$14x+4-9x^{2}=-1$$

Add 1 to both sides.

$$14x+4-9x^{2}+1=0$$

Add 4 and 1 to get 5.

$$14x+5-9x^{2}=0$$

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.

$$-9x^{2}+14x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 14 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-9\right)\times 5}}{2\left(-9\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-9\right)\times 5}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-14±\sqrt{196+36\times 5}}{2\left(-9\right)}$$

Multiply 36 times 5.

$$x=\frac{-14±\sqrt{196+180}}{2\left(-9\right)}$$

Add 196 to 180.

$$x=\frac{-14±\sqrt{376}}{2\left(-9\right)}$$

Take the square root of 376.

$$x=\frac{-14±2\sqrt{94}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-14±2\sqrt{94}}{-18}$$

Now solve the equation x=\frac{-14±2\sqrt{94}}{-18} when ± is plus. Add -14 to 2\sqrt{94}.

$$x=\frac{2\sqrt{94}-14}{-18}$$

Divide -14+2\sqrt{94} by -18.

$$x=\frac{7-\sqrt{94}}{9}$$

Now solve the equation x=\frac{-14±2\sqrt{94}}{-18} when ± is minus. Subtract 2\sqrt{94} from -14.

$$x=\frac{-2\sqrt{94}-14}{-18}$$

Divide -14-2\sqrt{94} by -18.

$$x=\frac{\sqrt{94}+7}{9}$$

The equation is now solved.

$$x=\frac{7-\sqrt{94}}{9} x=\frac{\sqrt{94}+7}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 3x+2 by 2.

$$6x+4=\left(x-1\right)\left(9x+1\right)$$

Use the distributive property to multiply x-1 by 9x+1 and combine like terms.

$$6x+4=9x^{2}-8x-1$$

Subtract 9x^{2} from both sides.

$$6x+4-9x^{2}=-8x-1$$

Add 8x to both sides.

$$6x+4-9x^{2}+8x=-1$$

Combine 6x and 8x to get 14x.

$$14x+4-9x^{2}=-1$$

Subtract 4 from both sides.

$$14x-9x^{2}=-1-4$$

Subtract 4 from -1 to get -5.

$$14x-9x^{2}=-5$$

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.

$$-9x^{2}+14x=-5$$

Divide both sides by -9.

$$\frac{-9x^{2}+14x}{-9}=-\frac{5}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{14}{-9}x=-\frac{5}{-9}$$

Divide 14 by -9.

$$x^{2}-\frac{14}{9}x=-\frac{5}{-9}$$

Divide -5 by -9.

$$x^{2}-\frac{14}{9}x=\frac{5}{9}$$

Divide -\frac{14}{9}, the coefficient of the x term, by 2 to get -\frac{7}{9}. Then add the square of -\frac{7}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{9}x+\left(-\frac{7}{9}\right)^{2}=\frac{5}{9}+\left(-\frac{7}{9}\right)^{2}$$

Square -\frac{7}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{9}x+\frac{49}{81}=\frac{5}{9}+\frac{49}{81}$$

Add \frac{5}{9} to \frac{49}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{9}x+\frac{49}{81}=\frac{94}{81}$$

Factor x^{2}-\frac{14}{9}x+\frac{49}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{9}\right)^{2}=\frac{94}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{9}\right)^{2}}=\sqrt{\frac{94}{81}}$$

Simplify.

$$x-\frac{7}{9}=\frac{\sqrt{94}}{9} x-\frac{7}{9}=-\frac{\sqrt{94}}{9}$$

Add \frac{7}{9} to both sides of the equation.

$$x=\frac{\sqrt{94}+7}{9} x=\frac{7-\sqrt{94}}{9}$$