Solve (6x-5)^2-(8x+9)^2=0

$$( 6 x - 5 ) ^ { 2 } - ( 8 x + 9 ) ^ { 2 } = 0$$

Solve for x

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

$$x=-7$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-60x+25-\left(8x+9\right)^{2}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8x+9\right)^{2}.

$$36x^{2}-60x+25-\left(64x^{2}+144x+81\right)=0$$

To find the opposite of 64x^{2}+144x+81, find the opposite of each term.

$$36x^{2}-60x+25-64x^{2}-144x-81=0$$

Combine 36x^{2} and -64x^{2} to get -28x^{2}.

$$-28x^{2}-60x+25-144x-81=0$$

Combine -60x and -144x to get -204x.

$$-28x^{2}-204x+25-81=0$$

Subtract 81 from 25 to get -56.

$$-28x^{2}-204x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -28 for a, -204 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-204\right)±\sqrt{\left(-204\right)^{2}-4\left(-28\right)\left(-56\right)}}{2\left(-28\right)}$$

Square -204.

$$x=\frac{-\left(-204\right)±\sqrt{41616-4\left(-28\right)\left(-56\right)}}{2\left(-28\right)}$$

Multiply -4 times -28.

$$x=\frac{-\left(-204\right)±\sqrt{41616+112\left(-56\right)}}{2\left(-28\right)}$$

Multiply 112 times -56.

$$x=\frac{-\left(-204\right)±\sqrt{41616-6272}}{2\left(-28\right)}$$

Add 41616 to -6272.

$$x=\frac{-\left(-204\right)±\sqrt{35344}}{2\left(-28\right)}$$

Take the square root of 35344.

$$x=\frac{-\left(-204\right)±188}{2\left(-28\right)}$$

The opposite of -204 is 204.

$$x=\frac{204±188}{2\left(-28\right)}$$

Multiply 2 times -28.

$$x=\frac{204±188}{-56}$$

Now solve the equation x=\frac{204±188}{-56} when ± is plus. Add 204 to 188.

$$x=\frac{392}{-56}$$

Divide 392 by -56.

$$x=-7$$

Now solve the equation x=\frac{204±188}{-56} when ± is minus. Subtract 188 from 204.

$$x=\frac{16}{-56}$$

Reduce the fraction \frac{16}{-56} to lowest terms by extracting and canceling out 8.

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

The equation is now solved.

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

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-5\right)^{2}.

$$36x^{2}-60x+25-\left(8x+9\right)^{2}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8x+9\right)^{2}.

$$36x^{2}-60x+25-\left(64x^{2}+144x+81\right)=0$$

To find the opposite of 64x^{2}+144x+81, find the opposite of each term.

$$36x^{2}-60x+25-64x^{2}-144x-81=0$$

Combine 36x^{2} and -64x^{2} to get -28x^{2}.

$$-28x^{2}-60x+25-144x-81=0$$

Combine -60x and -144x to get -204x.

$$-28x^{2}-204x+25-81=0$$

Subtract 81 from 25 to get -56.

$$-28x^{2}-204x-56=0$$

Add 56 to both sides. Anything plus zero gives itself.

$$-28x^{2}-204x=56$$

Divide both sides by -28.

$$\frac{-28x^{2}-204x}{-28}=\frac{56}{-28}$$

Dividing by -28 undoes the multiplication by -28.

$$x^{2}+\left(-\frac{204}{-28}\right)x=\frac{56}{-28}$$

Reduce the fraction \frac{-204}{-28} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{51}{7}x=\frac{56}{-28}$$

Divide 56 by -28.

$$x^{2}+\frac{51}{7}x=-2$$

Divide \frac{51}{7}, the coefficient of the x term, by 2 to get \frac{51}{14}. Then add the square of \frac{51}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{51}{7}x+\left(\frac{51}{14}\right)^{2}=-2+\left(\frac{51}{14}\right)^{2}$$

Square \frac{51}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{51}{7}x+\frac{2601}{196}=-2+\frac{2601}{196}$$

Add -2 to \frac{2601}{196}.

$$x^{2}+\frac{51}{7}x+\frac{2601}{196}=\frac{2209}{196}$$

Factor x^{2}+\frac{51}{7}x+\frac{2601}{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}.

$$\left(x+\frac{51}{14}\right)^{2}=\frac{2209}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{51}{14}\right)^{2}}=\sqrt{\frac{2209}{196}}$$

Simplify.

$$x+\frac{51}{14}=\frac{47}{14} x+\frac{51}{14}=-\frac{47}{14}$$

Subtract \frac{51}{14} from both sides of the equation.

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