Solve (9)(x-1)^2+2(x-1)-3=0

$$( 9 ) ( x - 1 ) ^ { 2 } + 2 ( x - 1 ) - 3 = 0$$

Solve for x

$$x = \frac{2 \sqrt{7} + 8}{9} \approx 1.476833625$$

$$x=\frac{8-2\sqrt{7}}{9}\approx 0.300944153$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

Use the distributive property to multiply 2 by x-1.

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

Combine -18x and 2x to get -16x.

$$9x^{2}-16x+9-2-3=0$$

Subtract 2 from 9 to get 7.

$$9x^{2}-16x+7-3=0$$

Subtract 3 from 7 to get 4.

$$9x^{2}-16x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -16 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 9\times 4}}{2\times 9}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 9\times 4}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-16\right)±\sqrt{256-36\times 4}}{2\times 9}$$

Multiply -36 times 4.

$$x=\frac{-\left(-16\right)±\sqrt{256-144}}{2\times 9}$$

Add 256 to -144.

$$x=\frac{-\left(-16\right)±\sqrt{112}}{2\times 9}$$

Take the square root of 112.

$$x=\frac{-\left(-16\right)±4\sqrt{7}}{2\times 9}$$

The opposite of -16 is 16.

$$x=\frac{16±4\sqrt{7}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{16±4\sqrt{7}}{18}$$

Now solve the equation x=\frac{16±4\sqrt{7}}{18} when ± is plus. Add 16 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+16}{18}$$

Divide 16+4\sqrt{7} by 18.

$$x=\frac{2\sqrt{7}+8}{9}$$

Now solve the equation x=\frac{16±4\sqrt{7}}{18} when ± is minus. Subtract 4\sqrt{7} from 16.

$$x=\frac{16-4\sqrt{7}}{18}$$

Divide 16-4\sqrt{7} by 18.

$$x=\frac{8-2\sqrt{7}}{9}$$

The equation is now solved.

$$x=\frac{2\sqrt{7}+8}{9} x=\frac{8-2\sqrt{7}}{9}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

Use the distributive property to multiply 2 by x-1.

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

Combine -18x and 2x to get -16x.

$$9x^{2}-16x+9-2-3=0$$

Subtract 2 from 9 to get 7.

$$9x^{2}-16x+7-3=0$$

Subtract 3 from 7 to get 4.

$$9x^{2}-16x+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$9x^{2}-16x=-4$$

Divide both sides by 9.

$$\frac{9x^{2}-16x}{9}=-\frac{4}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{16}{9}x=-\frac{4}{9}$$

Divide -\frac{16}{9}, the coefficient of the x term, by 2 to get -\frac{8}{9}. Then add the square of -\frac{8}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{9}x+\left(-\frac{8}{9}\right)^{2}=-\frac{4}{9}+\left(-\frac{8}{9}\right)^{2}$$

Square -\frac{8}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{9}x+\frac{64}{81}=-\frac{4}{9}+\frac{64}{81}$$

Add -\frac{4}{9} to \frac{64}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{9}x+\frac{64}{81}=\frac{28}{81}$$

Factor x^{2}-\frac{16}{9}x+\frac{64}{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}.

$$\left(x-\frac{8}{9}\right)^{2}=\frac{28}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{9}\right)^{2}}=\sqrt{\frac{28}{81}}$$

Simplify.

$$x-\frac{8}{9}=\frac{2\sqrt{7}}{9} x-\frac{8}{9}=-\frac{2\sqrt{7}}{9}$$

Add \frac{8}{9} to both sides of the equation.

$$x=\frac{2\sqrt{7}+8}{9} x=\frac{8-2\sqrt{7}}{9}$$