Solve 9(x-1)^2-16=0

$$9 ( x - 1 ) ^ { 2 } - 16 = 0$$

Solve for x

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

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

Steps Using Factoring By Grouping

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

$$9\left(x^{2}-2x+1\right)-16=0$$

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

$$9x^{2}-18x+9-16=0$$

Subtract 16 from 9 to get -7.

$$9x^{2}-18x-7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=-18 ab=9\left(-7\right)=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -63.

$$1,-63 3,-21 7,-9$$

Calculate the sum for each pair.

$$1-63=-62 3-21=-18 7-9=-2$$

The solution is the pair that gives sum -18.

$$a=-21 b=3$$

Rewrite 9x^{2}-18x-7 as \left(9x^{2}-21x\right)+\left(3x-7\right).

$$\left(9x^{2}-21x\right)+\left(3x-7\right)$$

Factor out 3x in 9x^{2}-21x.

$$3x\left(3x-7\right)+3x-7$$

Factor out common term 3x-7 by using distributive property.

$$\left(3x-7\right)\left(3x+1\right)$$

To find equation solutions, solve 3x-7=0 and 3x+1=0.

$$x=\frac{7}{3} x=-\frac{1}{3}$$

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}.

$$9\left(x^{2}-2x+1\right)-16=0$$

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

$$9x^{2}-18x+9-16=0$$

Subtract 16 from 9 to get -7.

$$9x^{2}-18x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -18 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 9\left(-7\right)}}{2\times 9}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 9\left(-7\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-18\right)±\sqrt{324-36\left(-7\right)}}{2\times 9}$$

Multiply -36 times -7.

$$x=\frac{-\left(-18\right)±\sqrt{324+252}}{2\times 9}$$

Add 324 to 252.

$$x=\frac{-\left(-18\right)±\sqrt{576}}{2\times 9}$$

Take the square root of 576.

$$x=\frac{-\left(-18\right)±24}{2\times 9}$$

The opposite of -18 is 18.

$$x=\frac{18±24}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{18±24}{18}$$

Now solve the equation x=\frac{18±24}{18} when ± is plus. Add 18 to 24.

$$x=\frac{42}{18}$$

Reduce the fraction \frac{42}{18} to lowest terms by extracting and canceling out 6.

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{18±24}{18} when ± is minus. Subtract 24 from 18.

$$x=-\frac{6}{18}$$

Reduce the fraction \frac{-6}{18} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=\frac{7}{3} x=-\frac{1}{3}$$

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}.

$$9\left(x^{2}-2x+1\right)-16=0$$

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

$$9x^{2}-18x+9-16=0$$

Subtract 16 from 9 to get -7.

$$9x^{2}-18x-7=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$9x^{2}-18x=7$$

Divide both sides by 9.

$$\frac{9x^{2}-18x}{9}=\frac{7}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{18}{9}\right)x=\frac{7}{9}$$

Divide -18 by 9.

$$x^{2}-2x=\frac{7}{9}$$

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

$$x^{2}-2x+1=\frac{7}{9}+1$$

Add \frac{7}{9} to 1.

$$x^{2}-2x+1=\frac{16}{9}$$

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

$$\left(x-1\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

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

Add 1 to both sides of the equation.

$$x=\frac{7}{3} x=-\frac{1}{3}$$