Solve (2x-5)(12x+5)=(3+10x)^2+11(2x-1)^2

$$( 2 x - 5 ) ( 12 x + 5 ) = ( 3 + 10 x ) ^ { 2 } + 11 ( 2 x - 1 ) ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{-11+\sqrt{479}i}{40}\approx -0.275+0.547151716i$$

$$x=\frac{-\sqrt{479}i-11}{40}\approx -0.275-0.547151716i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-5 by 12x+5 and combine like terms.

$$24x^{2}-50x-25=\left(3+10x\right)^{2}+11\left(2x-1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+10x\right)^{2}.

$$24x^{2}-50x-25=9+60x+100x^{2}+11\left(2x-1\right)^{2}$$

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

$$24x^{2}-50x-25=9+60x+100x^{2}+11\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 11 by 4x^{2}-4x+1.

$$24x^{2}-50x-25=9+60x+100x^{2}+44x^{2}-44x+11$$

Combine 100x^{2} and 44x^{2} to get 144x^{2}.

$$24x^{2}-50x-25=9+60x+144x^{2}-44x+11$$

Combine 60x and -44x to get 16x.

$$24x^{2}-50x-25=9+16x+144x^{2}+11$$

Add 9 and 11 to get 20.

$$24x^{2}-50x-25=20+16x+144x^{2}$$

Subtract 20 from both sides.

$$24x^{2}-50x-25-20=16x+144x^{2}$$

Subtract 20 from -25 to get -45.

$$24x^{2}-50x-45=16x+144x^{2}$$

Subtract 16x from both sides.

$$24x^{2}-50x-45-16x=144x^{2}$$

Combine -50x and -16x to get -66x.

$$24x^{2}-66x-45=144x^{2}$$

Subtract 144x^{2} from both sides.

$$24x^{2}-66x-45-144x^{2}=0$$

Combine 24x^{2} and -144x^{2} to get -120x^{2}.

$$-120x^{2}-66x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -120 for a, -66 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\left(-120\right)\left(-45\right)}}{2\left(-120\right)}$$

Square -66.

$$x=\frac{-\left(-66\right)±\sqrt{4356-4\left(-120\right)\left(-45\right)}}{2\left(-120\right)}$$

Multiply -4 times -120.

$$x=\frac{-\left(-66\right)±\sqrt{4356+480\left(-45\right)}}{2\left(-120\right)}$$

Multiply 480 times -45.

$$x=\frac{-\left(-66\right)±\sqrt{4356-21600}}{2\left(-120\right)}$$

Add 4356 to -21600.

$$x=\frac{-\left(-66\right)±\sqrt{-17244}}{2\left(-120\right)}$$

Take the square root of -17244.

$$x=\frac{-\left(-66\right)±6\sqrt{479}i}{2\left(-120\right)}$$

The opposite of -66 is 66.

$$x=\frac{66±6\sqrt{479}i}{2\left(-120\right)}$$

Multiply 2 times -120.

$$x=\frac{66±6\sqrt{479}i}{-240}$$

Now solve the equation x=\frac{66±6\sqrt{479}i}{-240} when ± is plus. Add 66 to 6i\sqrt{479}.

$$x=\frac{66+6\sqrt{479}i}{-240}$$

Divide 66+6i\sqrt{479} by -240.

$$x=\frac{-\sqrt{479}i-11}{40}$$

Now solve the equation x=\frac{66±6\sqrt{479}i}{-240} when ± is minus. Subtract 6i\sqrt{479} from 66.

$$x=\frac{-6\sqrt{479}i+66}{-240}$$

Divide 66-6i\sqrt{479} by -240.

$$x=\frac{-11+\sqrt{479}i}{40}$$

The equation is now solved.

$$x=\frac{-\sqrt{479}i-11}{40} x=\frac{-11+\sqrt{479}i}{40}$$

Steps for Completing the Square

Use the distributive property to multiply 2x-5 by 12x+5 and combine like terms.

$$24x^{2}-50x-25=\left(3+10x\right)^{2}+11\left(2x-1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+10x\right)^{2}.

$$24x^{2}-50x-25=9+60x+100x^{2}+11\left(2x-1\right)^{2}$$

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

$$24x^{2}-50x-25=9+60x+100x^{2}+11\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 11 by 4x^{2}-4x+1.

$$24x^{2}-50x-25=9+60x+100x^{2}+44x^{2}-44x+11$$

Combine 100x^{2} and 44x^{2} to get 144x^{2}.

$$24x^{2}-50x-25=9+60x+144x^{2}-44x+11$$

Combine 60x and -44x to get 16x.

$$24x^{2}-50x-25=9+16x+144x^{2}+11$$

Add 9 and 11 to get 20.

$$24x^{2}-50x-25=20+16x+144x^{2}$$

Subtract 16x from both sides.

$$24x^{2}-50x-25-16x=20+144x^{2}$$

Combine -50x and -16x to get -66x.

$$24x^{2}-66x-25=20+144x^{2}$$

Subtract 144x^{2} from both sides.

$$24x^{2}-66x-25-144x^{2}=20$$

Combine 24x^{2} and -144x^{2} to get -120x^{2}.

$$-120x^{2}-66x-25=20$$

Add 25 to both sides.

$$-120x^{2}-66x=20+25$$

Add 20 and 25 to get 45.

$$-120x^{2}-66x=45$$

Divide both sides by -120.

$$\frac{-120x^{2}-66x}{-120}=\frac{45}{-120}$$

Dividing by -120 undoes the multiplication by -120.

$$x^{2}+\left(-\frac{66}{-120}\right)x=\frac{45}{-120}$$

Reduce the fraction \frac{-66}{-120} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{11}{20}x=\frac{45}{-120}$$

Reduce the fraction \frac{45}{-120} to lowest terms by extracting and canceling out 15.

$$x^{2}+\frac{11}{20}x=-\frac{3}{8}$$

Divide \frac{11}{20}, the coefficient of the x term, by 2 to get \frac{11}{40}. Then add the square of \frac{11}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{20}x+\left(\frac{11}{40}\right)^{2}=-\frac{3}{8}+\left(\frac{11}{40}\right)^{2}$$

Square \frac{11}{40} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{20}x+\frac{121}{1600}=-\frac{3}{8}+\frac{121}{1600}$$

Add -\frac{3}{8} to \frac{121}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{20}x+\frac{121}{1600}=-\frac{479}{1600}$$

Factor x^{2}+\frac{11}{20}x+\frac{121}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{40}\right)^{2}=-\frac{479}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{40}\right)^{2}}=\sqrt{-\frac{479}{1600}}$$

Simplify.

$$x+\frac{11}{40}=\frac{\sqrt{479}i}{40} x+\frac{11}{40}=-\frac{\sqrt{479}i}{40}$$

Subtract \frac{11}{40} from both sides of the equation.

$$x=\frac{-11+\sqrt{479}i}{40} x=\frac{-\sqrt{479}i-11}{40}$$