Solve 11(5x-6)^2=(5x+12)^2

$$11 ( 5 x - 6 ) ^ { 2 } = ( 5 x + 12 ) ^ { 2 }$$

Solve for x

$$x = \frac{9 \sqrt{11} + 39}{25} \approx 2.753984925$$

$$x=\frac{39-9\sqrt{11}}{25}\approx 0.366015075$$

Steps Using the Quadratic Formula

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

$$11\left(25x^{2}-60x+36\right)=\left(5x+12\right)^{2}$$

Use the distributive property to multiply 11 by 25x^{2}-60x+36.

$$275x^{2}-660x+396=\left(5x+12\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+12\right)^{2}.

$$275x^{2}-660x+396=25x^{2}+120x+144$$

Subtract 25x^{2} from both sides.

$$275x^{2}-660x+396-25x^{2}=120x+144$$

Combine 275x^{2} and -25x^{2} to get 250x^{2}.

$$250x^{2}-660x+396=120x+144$$

Subtract 120x from both sides.

$$250x^{2}-660x+396-120x=144$$

Combine -660x and -120x to get -780x.

$$250x^{2}-780x+396=144$$

Subtract 144 from both sides.

$$250x^{2}-780x+396-144=0$$

Subtract 144 from 396 to get 252.

$$250x^{2}-780x+252=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 250 for a, -780 for b, and 252 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-780\right)±\sqrt{\left(-780\right)^{2}-4\times 250\times 252}}{2\times 250}$$

Square -780.

$$x=\frac{-\left(-780\right)±\sqrt{608400-4\times 250\times 252}}{2\times 250}$$

Multiply -4 times 250.

$$x=\frac{-\left(-780\right)±\sqrt{608400-1000\times 252}}{2\times 250}$$

Multiply -1000 times 252.

$$x=\frac{-\left(-780\right)±\sqrt{608400-252000}}{2\times 250}$$

Add 608400 to -252000.

$$x=\frac{-\left(-780\right)±\sqrt{356400}}{2\times 250}$$

Take the square root of 356400.

$$x=\frac{-\left(-780\right)±180\sqrt{11}}{2\times 250}$$

The opposite of -780 is 780.

$$x=\frac{780±180\sqrt{11}}{2\times 250}$$

Multiply 2 times 250.

$$x=\frac{780±180\sqrt{11}}{500}$$

Now solve the equation x=\frac{780±180\sqrt{11}}{500} when ± is plus. Add 780 to 180\sqrt{11}.

$$x=\frac{180\sqrt{11}+780}{500}$$

Divide 780+180\sqrt{11} by 500.

$$x=\frac{9\sqrt{11}+39}{25}$$

Now solve the equation x=\frac{780±180\sqrt{11}}{500} when ± is minus. Subtract 180\sqrt{11} from 780.

$$x=\frac{780-180\sqrt{11}}{500}$$

Divide 780-180\sqrt{11} by 500.

$$x=\frac{39-9\sqrt{11}}{25}$$

The equation is now solved.

$$x=\frac{9\sqrt{11}+39}{25} x=\frac{39-9\sqrt{11}}{25}$$

Steps for Completing the Square

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

$$11\left(25x^{2}-60x+36\right)=\left(5x+12\right)^{2}$$

Use the distributive property to multiply 11 by 25x^{2}-60x+36.

$$275x^{2}-660x+396=\left(5x+12\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+12\right)^{2}.

$$275x^{2}-660x+396=25x^{2}+120x+144$$

Subtract 25x^{2} from both sides.

$$275x^{2}-660x+396-25x^{2}=120x+144$$

Combine 275x^{2} and -25x^{2} to get 250x^{2}.

$$250x^{2}-660x+396=120x+144$$

Subtract 120x from both sides.

$$250x^{2}-660x+396-120x=144$$

Combine -660x and -120x to get -780x.

$$250x^{2}-780x+396=144$$

Subtract 396 from both sides.

$$250x^{2}-780x=144-396$$

Subtract 396 from 144 to get -252.

$$250x^{2}-780x=-252$$

Divide both sides by 250.

$$\frac{250x^{2}-780x}{250}=-\frac{252}{250}$$

Dividing by 250 undoes the multiplication by 250.

$$x^{2}+\left(-\frac{780}{250}\right)x=-\frac{252}{250}$$

Reduce the fraction \frac{-780}{250} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{78}{25}x=-\frac{252}{250}$$

Reduce the fraction \frac{-252}{250} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{78}{25}x=-\frac{126}{125}$$

Divide -\frac{78}{25}, the coefficient of the x term, by 2 to get -\frac{39}{25}. Then add the square of -\frac{39}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{78}{25}x+\left(-\frac{39}{25}\right)^{2}=-\frac{126}{125}+\left(-\frac{39}{25}\right)^{2}$$

Square -\frac{39}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{78}{25}x+\frac{1521}{625}=-\frac{126}{125}+\frac{1521}{625}$$

Add -\frac{126}{125} to \frac{1521}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{78}{25}x+\frac{1521}{625}=\frac{891}{625}$$

Factor x^{2}-\frac{78}{25}x+\frac{1521}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{39}{25}\right)^{2}=\frac{891}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{25}\right)^{2}}=\sqrt{\frac{891}{625}}$$

Simplify.

$$x-\frac{39}{25}=\frac{9\sqrt{11}}{25} x-\frac{39}{25}=-\frac{9\sqrt{11}}{25}$$

Add \frac{39}{25} to both sides of the equation.

$$x=\frac{9\sqrt{11}+39}{25} x=\frac{39-9\sqrt{11}}{25}$$