Solve 144=(-10-(100+40x+4x^2))+(-15-(625+50x+x^2))+(68-(7744+352x+4x^2))

$$144=(-10-(100+40x+4 { x }^{ 2 } ))+(-15-(625+50x+ { x }^{ 2 } ))+(68-(7744+352x+4 { x }^{ 2 } ))$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{28289}i-221}{9}\approx -24.555555556-18.688149014i$$

$$x=\frac{-221+\sqrt{28289}i}{9}\approx -24.555555556+18.688149014i$$

Steps Using the Quadratic Formula

To find the opposite of 100+40x+4x^{2}, find the opposite of each term.

$$144=-10-100-40x-4x^{2}-15-\left(625+50x+x^{2}\right)+68-\left(7744+352x+4x^{2}\right)$$

Subtract 100 from -10 to get -110.

$$144=-110-40x-4x^{2}-15-\left(625+50x+x^{2}\right)+68-\left(7744+352x+4x^{2}\right)$$

Subtract 15 from -110 to get -125.

$$144=-125-40x-4x^{2}-\left(625+50x+x^{2}\right)+68-\left(7744+352x+4x^{2}\right)$$

To find the opposite of 625+50x+x^{2}, find the opposite of each term.

$$144=-125-40x-4x^{2}-625-50x-x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Subtract 625 from -125 to get -750.

$$144=-750-40x-4x^{2}-50x-x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Combine -40x and -50x to get -90x.

$$144=-750-90x-4x^{2}-x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$144=-750-90x-5x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Add -750 and 68 to get -682.

$$144=-682-90x-5x^{2}-\left(7744+352x+4x^{2}\right)$$

To find the opposite of 7744+352x+4x^{2}, find the opposite of each term.

$$144=-682-90x-5x^{2}-7744-352x-4x^{2}$$

Subtract 7744 from -682 to get -8426.

$$144=-8426-90x-5x^{2}-352x-4x^{2}$$

Combine -90x and -352x to get -442x.

$$144=-8426-442x-5x^{2}-4x^{2}$$

Combine -5x^{2} and -4x^{2} to get -9x^{2}.

$$144=-8426-442x-9x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-8426-442x-9x^{2}=144$$

Subtract 144 from both sides.

$$-8426-442x-9x^{2}-144=0$$

Subtract 144 from -8426 to get -8570.

$$-8570-442x-9x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-9x^{2}-442x-8570=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -442 for b, and -8570 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-442\right)±\sqrt{\left(-442\right)^{2}-4\left(-9\right)\left(-8570\right)}}{2\left(-9\right)}$$

Square -442.

$$x=\frac{-\left(-442\right)±\sqrt{195364-4\left(-9\right)\left(-8570\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-442\right)±\sqrt{195364+36\left(-8570\right)}}{2\left(-9\right)}$$

Multiply 36 times -8570.

$$x=\frac{-\left(-442\right)±\sqrt{195364-308520}}{2\left(-9\right)}$$

Add 195364 to -308520.

$$x=\frac{-\left(-442\right)±\sqrt{-113156}}{2\left(-9\right)}$$

Take the square root of -113156.

$$x=\frac{-\left(-442\right)±2\sqrt{28289}i}{2\left(-9\right)}$$

The opposite of -442 is 442.

$$x=\frac{442±2\sqrt{28289}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{442±2\sqrt{28289}i}{-18}$$

Now solve the equation x=\frac{442±2\sqrt{28289}i}{-18} when ± is plus. Add 442 to 2i\sqrt{28289}.

$$x=\frac{442+2\sqrt{28289}i}{-18}$$

Divide 442+2i\sqrt{28289} by -18.

$$x=\frac{-\sqrt{28289}i-221}{9}$$

Now solve the equation x=\frac{442±2\sqrt{28289}i}{-18} when ± is minus. Subtract 2i\sqrt{28289} from 442.

$$x=\frac{-2\sqrt{28289}i+442}{-18}$$

Divide 442-2i\sqrt{28289} by -18.

$$x=\frac{-221+\sqrt{28289}i}{9}$$

The equation is now solved.

$$x=\frac{-\sqrt{28289}i-221}{9} x=\frac{-221+\sqrt{28289}i}{9}$$

Steps for Completing the Square

To find the opposite of 100+40x+4x^{2}, find the opposite of each term.

$$144=-10-100-40x-4x^{2}-15-\left(625+50x+x^{2}\right)+68-\left(7744+352x+4x^{2}\right)$$

Subtract 100 from -10 to get -110.

$$144=-110-40x-4x^{2}-15-\left(625+50x+x^{2}\right)+68-\left(7744+352x+4x^{2}\right)$$

Subtract 15 from -110 to get -125.

$$144=-125-40x-4x^{2}-\left(625+50x+x^{2}\right)+68-\left(7744+352x+4x^{2}\right)$$

To find the opposite of 625+50x+x^{2}, find the opposite of each term.

$$144=-125-40x-4x^{2}-625-50x-x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Subtract 625 from -125 to get -750.

$$144=-750-40x-4x^{2}-50x-x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Combine -40x and -50x to get -90x.

$$144=-750-90x-4x^{2}-x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$144=-750-90x-5x^{2}+68-\left(7744+352x+4x^{2}\right)$$

Add -750 and 68 to get -682.

$$144=-682-90x-5x^{2}-\left(7744+352x+4x^{2}\right)$$

To find the opposite of 7744+352x+4x^{2}, find the opposite of each term.

$$144=-682-90x-5x^{2}-7744-352x-4x^{2}$$

Subtract 7744 from -682 to get -8426.

$$144=-8426-90x-5x^{2}-352x-4x^{2}$$

Combine -90x and -352x to get -442x.

$$144=-8426-442x-5x^{2}-4x^{2}$$

Combine -5x^{2} and -4x^{2} to get -9x^{2}.

$$144=-8426-442x-9x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-8426-442x-9x^{2}=144$$

Add 8426 to both sides.

$$-442x-9x^{2}=144+8426$$

Add 144 and 8426 to get 8570.

$$-442x-9x^{2}=8570$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-9x^{2}-442x=8570$$

Divide both sides by -9.

$$\frac{-9x^{2}-442x}{-9}=\frac{8570}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{442}{-9}\right)x=\frac{8570}{-9}$$

Divide -442 by -9.

$$x^{2}+\frac{442}{9}x=\frac{8570}{-9}$$

Divide 8570 by -9.

$$x^{2}+\frac{442}{9}x=-\frac{8570}{9}$$

Divide \frac{442}{9}, the coefficient of the x term, by 2 to get \frac{221}{9}. Then add the square of \frac{221}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{442}{9}x+\left(\frac{221}{9}\right)^{2}=-\frac{8570}{9}+\left(\frac{221}{9}\right)^{2}$$

Square \frac{221}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{442}{9}x+\frac{48841}{81}=-\frac{8570}{9}+\frac{48841}{81}$$

Add -\frac{8570}{9} to \frac{48841}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{442}{9}x+\frac{48841}{81}=-\frac{28289}{81}$$

Factor x^{2}+\frac{442}{9}x+\frac{48841}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{221}{9}\right)^{2}=-\frac{28289}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{221}{9}\right)^{2}}=\sqrt{-\frac{28289}{81}}$$

Simplify.

$$x+\frac{221}{9}=\frac{\sqrt{28289}i}{9} x+\frac{221}{9}=-\frac{\sqrt{28289}i}{9}$$

Subtract \frac{221}{9} from both sides of the equation.

$$x=\frac{-221+\sqrt{28289}i}{9} x=\frac{-\sqrt{28289}i-221}{9}$$