Solve 5000-150x-x^2=4516

$$5000-150x- { x }^{ 2 } =4516$$

Solve for x (complex solution)

$$x=\sqrt{6109}-75\approx 3.160092119$$

$$x=-\left(\sqrt{6109}+75\right)\approx -153.160092119$$

Steps Using the Quadratic Formula

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.

$$-x^{2}-150x+5000=4516$$

Subtract 4516 from both sides of the equation.

$$-x^{2}-150x+5000-4516=4516-4516$$

Subtracting 4516 from itself leaves 0.

$$-x^{2}-150x+5000-4516=0$$

Subtract 4516 from 5000.

$$-x^{2}-150x+484=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -150 for b, and 484 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-150\right)±\sqrt{\left(-150\right)^{2}-4\left(-1\right)\times 484}}{2\left(-1\right)}$$

Square -150.

$$x=\frac{-\left(-150\right)±\sqrt{22500-4\left(-1\right)\times 484}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-150\right)±\sqrt{22500+4\times 484}}{2\left(-1\right)}$$

Multiply 4 times 484.

$$x=\frac{-\left(-150\right)±\sqrt{22500+1936}}{2\left(-1\right)}$$

Add 22500 to 1936.

$$x=\frac{-\left(-150\right)±\sqrt{24436}}{2\left(-1\right)}$$

Take the square root of 24436.

$$x=\frac{-\left(-150\right)±2\sqrt{6109}}{2\left(-1\right)}$$

The opposite of -150 is 150.

$$x=\frac{150±2\sqrt{6109}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{150±2\sqrt{6109}}{-2}$$

Now solve the equation x=\frac{150±2\sqrt{6109}}{-2} when ± is plus. Add 150 to 2\sqrt{6109}.

$$x=\frac{2\sqrt{6109}+150}{-2}$$

Divide 150+2\sqrt{6109} by -2.

$$x=-\left(\sqrt{6109}+75\right)$$

Now solve the equation x=\frac{150±2\sqrt{6109}}{-2} when ± is minus. Subtract 2\sqrt{6109} from 150.

$$x=\frac{150-2\sqrt{6109}}{-2}$$

Divide 150-2\sqrt{6109} by -2.

$$x=\sqrt{6109}-75$$

The equation is now solved.

$$x=-\left(\sqrt{6109}+75\right) x=\sqrt{6109}-75$$

Steps for Completing the Square

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.

$$-x^{2}-150x+5000=4516$$

Subtract 5000 from both sides of the equation.

$$-x^{2}-150x+5000-5000=4516-5000$$

Subtracting 5000 from itself leaves 0.

$$-x^{2}-150x=4516-5000$$

Subtract 5000 from 4516.

$$-x^{2}-150x=-484$$

Divide both sides by -1.

$$\frac{-x^{2}-150x}{-1}=-\frac{484}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{150}{-1}\right)x=-\frac{484}{-1}$$

Divide -150 by -1.

$$x^{2}+150x=-\frac{484}{-1}$$

Divide -484 by -1.

$$x^{2}+150x=484$$

Divide 150, the coefficient of the x term, by 2 to get 75. Then add the square of 75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+150x+75^{2}=484+75^{2}$$

Square 75.

$$x^{2}+150x+5625=484+5625$$

Add 484 to 5625.

$$x^{2}+150x+5625=6109$$

Factor x^{2}+150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+75\right)^{2}=6109$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+75\right)^{2}}=\sqrt{6109}$$

Simplify.

$$x+75=\sqrt{6109} x+75=-\sqrt{6109}$$

Subtract 75 from both sides of the equation.

$$x=\sqrt{6109}-75 x=-\sqrt{6109}-75$$

Solve for x

$$x=\sqrt{6109}-75\approx 3.160092119$$

$$x=-\sqrt{6109}-75\approx -153.160092119$$

Steps Using the Quadratic Formula

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.

$$-x^{2}-150x+5000=4516$$

Subtract 4516 from both sides of the equation.

$$-x^{2}-150x+5000-4516=4516-4516$$

Subtracting 4516 from itself leaves 0.

$$-x^{2}-150x+5000-4516=0$$

Subtract 4516 from 5000.

$$-x^{2}-150x+484=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -150 for b, and 484 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-150\right)±\sqrt{\left(-150\right)^{2}-4\left(-1\right)\times 484}}{2\left(-1\right)}$$

Square -150.

$$x=\frac{-\left(-150\right)±\sqrt{22500-4\left(-1\right)\times 484}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-150\right)±\sqrt{22500+4\times 484}}{2\left(-1\right)}$$

Multiply 4 times 484.

$$x=\frac{-\left(-150\right)±\sqrt{22500+1936}}{2\left(-1\right)}$$

Add 22500 to 1936.

$$x=\frac{-\left(-150\right)±\sqrt{24436}}{2\left(-1\right)}$$

Take the square root of 24436.

$$x=\frac{-\left(-150\right)±2\sqrt{6109}}{2\left(-1\right)}$$

The opposite of -150 is 150.

$$x=\frac{150±2\sqrt{6109}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{150±2\sqrt{6109}}{-2}$$

Now solve the equation x=\frac{150±2\sqrt{6109}}{-2} when ± is plus. Add 150 to 2\sqrt{6109}.

$$x=\frac{2\sqrt{6109}+150}{-2}$$

Divide 150+2\sqrt{6109} by -2.

$$x=-\left(\sqrt{6109}+75\right)$$

Now solve the equation x=\frac{150±2\sqrt{6109}}{-2} when ± is minus. Subtract 2\sqrt{6109} from 150.

$$x=\frac{150-2\sqrt{6109}}{-2}$$

Divide 150-2\sqrt{6109} by -2.

$$x=\sqrt{6109}-75$$

The equation is now solved.

$$x=-\left(\sqrt{6109}+75\right) x=\sqrt{6109}-75$$

Steps for Completing the Square

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.

$$-x^{2}-150x+5000=4516$$

Subtract 5000 from both sides of the equation.

$$-x^{2}-150x+5000-5000=4516-5000$$

Subtracting 5000 from itself leaves 0.

$$-x^{2}-150x=4516-5000$$

Subtract 5000 from 4516.

$$-x^{2}-150x=-484$$

Divide both sides by -1.

$$\frac{-x^{2}-150x}{-1}=-\frac{484}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{150}{-1}\right)x=-\frac{484}{-1}$$

Divide -150 by -1.

$$x^{2}+150x=-\frac{484}{-1}$$

Divide -484 by -1.

$$x^{2}+150x=484$$

Divide 150, the coefficient of the x term, by 2 to get 75. Then add the square of 75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+150x+75^{2}=484+75^{2}$$

Square 75.

$$x^{2}+150x+5625=484+5625$$

Add 484 to 5625.

$$x^{2}+150x+5625=6109$$

Factor x^{2}+150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+75\right)^{2}=6109$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+75\right)^{2}}=\sqrt{6109}$$

Simplify.

$$x+75=\sqrt{6109} x+75=-\sqrt{6109}$$

Subtract 75 from both sides of the equation.

$$x=\sqrt{6109}-75 x=-\sqrt{6109}-75$$