Solve 5x^2-8x(2+14x)=108-x

$$5 { x }^{ 2 } -8x(2+14x)=108-x$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{5111}i-15}{214}\approx -0.070093458-1.0022139i$$

$$x=\frac{-15+3\sqrt{5111}i}{214}\approx -0.070093458+1.0022139i$$

Steps Using the Quadratic Formula

Subtract 108 from both sides.

$$5x^{2}-8x\left(2+14x\right)-108=-x$$

Add x to both sides.

$$5x^{2}-8x\left(2+14x\right)-108+x=0$$

Use the distributive property to multiply -8x by 2+14x.

$$5x^{2}-16x-112x^{2}-108+x=0$$

Combine 5x^{2} and -112x^{2} to get -107x^{2}.

$$-107x^{2}-16x-108+x=0$$

Combine -16x and x to get -15x.

$$-107x^{2}-15x-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -107 for a, -15 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-107\right)\left(-108\right)}}{2\left(-107\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-107\right)\left(-108\right)}}{2\left(-107\right)}$$

Multiply -4 times -107.

$$x=\frac{-\left(-15\right)±\sqrt{225+428\left(-108\right)}}{2\left(-107\right)}$$

Multiply 428 times -108.

$$x=\frac{-\left(-15\right)±\sqrt{225-46224}}{2\left(-107\right)}$$

Add 225 to -46224.

$$x=\frac{-\left(-15\right)±\sqrt{-45999}}{2\left(-107\right)}$$

Take the square root of -45999.

$$x=\frac{-\left(-15\right)±3\sqrt{5111}i}{2\left(-107\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±3\sqrt{5111}i}{2\left(-107\right)}$$

Multiply 2 times -107.

$$x=\frac{15±3\sqrt{5111}i}{-214}$$

Now solve the equation x=\frac{15±3\sqrt{5111}i}{-214} when ± is plus. Add 15 to 3i\sqrt{5111}.

$$x=\frac{15+3\sqrt{5111}i}{-214}$$

Divide 15+3i\sqrt{5111} by -214.

$$x=\frac{-3\sqrt{5111}i-15}{214}$$

Now solve the equation x=\frac{15±3\sqrt{5111}i}{-214} when ± is minus. Subtract 3i\sqrt{5111} from 15.

$$x=\frac{-3\sqrt{5111}i+15}{-214}$$

Divide 15-3i\sqrt{5111} by -214.

$$x=\frac{-15+3\sqrt{5111}i}{214}$$

The equation is now solved.

$$x=\frac{-3\sqrt{5111}i-15}{214} x=\frac{-15+3\sqrt{5111}i}{214}$$

Steps for Completing the Square

Add x to both sides.

$$5x^{2}-8x\left(2+14x\right)+x=108$$

Use the distributive property to multiply -8x by 2+14x.

$$5x^{2}-16x-112x^{2}+x=108$$

Combine 5x^{2} and -112x^{2} to get -107x^{2}.

$$-107x^{2}-16x+x=108$$

Combine -16x and x to get -15x.

$$-107x^{2}-15x=108$$

Divide both sides by -107.

$$\frac{-107x^{2}-15x}{-107}=\frac{108}{-107}$$

Dividing by -107 undoes the multiplication by -107.

$$x^{2}+\left(-\frac{15}{-107}\right)x=\frac{108}{-107}$$

Divide -15 by -107.

$$x^{2}+\frac{15}{107}x=\frac{108}{-107}$$

Divide 108 by -107.

$$x^{2}+\frac{15}{107}x=-\frac{108}{107}$$

Divide \frac{15}{107}, the coefficient of the x term, by 2 to get \frac{15}{214}. Then add the square of \frac{15}{214} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{107}x+\left(\frac{15}{214}\right)^{2}=-\frac{108}{107}+\left(\frac{15}{214}\right)^{2}$$

Square \frac{15}{214} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{107}x+\frac{225}{45796}=-\frac{108}{107}+\frac{225}{45796}$$

Add -\frac{108}{107} to \frac{225}{45796} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{15}{107}x+\frac{225}{45796}=-\frac{45999}{45796}$$

Factor x^{2}+\frac{15}{107}x+\frac{225}{45796}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{214}\right)^{2}=-\frac{45999}{45796}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{214}\right)^{2}}=\sqrt{-\frac{45999}{45796}}$$

Simplify.

$$x+\frac{15}{214}=\frac{3\sqrt{5111}i}{214} x+\frac{15}{214}=-\frac{3\sqrt{5111}i}{214}$$

Subtract \frac{15}{214} from both sides of the equation.

$$x=\frac{-15+3\sqrt{5111}i}{214} x=\frac{-3\sqrt{5111}i-15}{214}$$