Solve (12x-7)^2=(2x+3)^2

$$( 12 x - 7 ) ^ { 2 } = ( 2 x + 3 ) ^ { 2 }$$

Solve for x

$$x=1$$

$$x=\frac{2}{7}\approx 0.285714286$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12x-7\right)^{2}.

$$144x^{2}-168x+49=\left(2x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$144x^{2}-168x+49=4x^{2}+12x+9$$

Subtract 4x^{2} from both sides.

$$144x^{2}-168x+49-4x^{2}=12x+9$$

Combine 144x^{2} and -4x^{2} to get 140x^{2}.

$$140x^{2}-168x+49=12x+9$$

Subtract 12x from both sides.

$$140x^{2}-168x+49-12x=9$$

Combine -168x and -12x to get -180x.

$$140x^{2}-180x+49=9$$

Subtract 9 from both sides.

$$140x^{2}-180x+49-9=0$$

Subtract 9 from 49 to get 40.

$$140x^{2}-180x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 140 for a, -180 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-180\right)±\sqrt{\left(-180\right)^{2}-4\times 140\times 40}}{2\times 140}$$

Square -180.

$$x=\frac{-\left(-180\right)±\sqrt{32400-4\times 140\times 40}}{2\times 140}$$

Multiply -4 times 140.

$$x=\frac{-\left(-180\right)±\sqrt{32400-560\times 40}}{2\times 140}$$

Multiply -560 times 40.

$$x=\frac{-\left(-180\right)±\sqrt{32400-22400}}{2\times 140}$$

Add 32400 to -22400.

$$x=\frac{-\left(-180\right)±\sqrt{10000}}{2\times 140}$$

Take the square root of 10000.

$$x=\frac{-\left(-180\right)±100}{2\times 140}$$

The opposite of -180 is 180.

$$x=\frac{180±100}{2\times 140}$$

Multiply 2 times 140.

$$x=\frac{180±100}{280}$$

Now solve the equation x=\frac{180±100}{280} when ± is plus. Add 180 to 100.

$$x=\frac{280}{280}$$

Divide 280 by 280.

$$x=1$$

Now solve the equation x=\frac{180±100}{280} when ± is minus. Subtract 100 from 180.

$$x=\frac{80}{280}$$

Reduce the fraction \frac{80}{280} to lowest terms by extracting and canceling out 40.

$$x=\frac{2}{7}$$

The equation is now solved.

$$x=1 x=\frac{2}{7}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12x-7\right)^{2}.

$$144x^{2}-168x+49=\left(2x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$144x^{2}-168x+49=4x^{2}+12x+9$$

Subtract 4x^{2} from both sides.

$$144x^{2}-168x+49-4x^{2}=12x+9$$

Combine 144x^{2} and -4x^{2} to get 140x^{2}.

$$140x^{2}-168x+49=12x+9$$

Subtract 12x from both sides.

$$140x^{2}-168x+49-12x=9$$

Combine -168x and -12x to get -180x.

$$140x^{2}-180x+49=9$$

Subtract 49 from both sides.

$$140x^{2}-180x=9-49$$

Subtract 49 from 9 to get -40.

$$140x^{2}-180x=-40$$

Divide both sides by 140.

$$\frac{140x^{2}-180x}{140}=-\frac{40}{140}$$

Dividing by 140 undoes the multiplication by 140.

$$x^{2}+\left(-\frac{180}{140}\right)x=-\frac{40}{140}$$

Reduce the fraction \frac{-180}{140} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{9}{7}x=-\frac{40}{140}$$

Reduce the fraction \frac{-40}{140} to lowest terms by extracting and canceling out 20.

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

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

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

Square -\frac{9}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{7}x+\frac{81}{196}=-\frac{2}{7}+\frac{81}{196}$$

Add -\frac{2}{7} to \frac{81}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{7}x+\frac{81}{196}=\frac{25}{196}$$

Factor x^{2}-\frac{9}{7}x+\frac{81}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{14}\right)^{2}=\frac{25}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{14}\right)^{2}}=\sqrt{\frac{25}{196}}$$

Simplify.

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

Add \frac{9}{14} to both sides of the equation.

$$x=1 x=\frac{2}{7}$$