Solve x^2+5000x-500000=0

$$x ^ { 2 } + 5000 x - 500000 = 0$$

Solve for x

$$x=1500\sqrt{3}-2500\approx 98.076211353$$

$$x=-1500\sqrt{3}-2500\approx -5098.076211353$$

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.

$$x^{2}+5000x-500000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5000 for b, and -500000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5000±\sqrt{5000^{2}-4\left(-500000\right)}}{2}$$

Square 5000.

$$x=\frac{-5000±\sqrt{25000000-4\left(-500000\right)}}{2}$$

Multiply -4 times -500000.

$$x=\frac{-5000±\sqrt{25000000+2000000}}{2}$$

Add 25000000 to 2000000.

$$x=\frac{-5000±\sqrt{27000000}}{2}$$

Take the square root of 27000000.

$$x=\frac{-5000±3000\sqrt{3}}{2}$$

Now solve the equation x=\frac{-5000±3000\sqrt{3}}{2} when ± is plus. Add -5000 to 3000\sqrt{3}.

$$x=\frac{3000\sqrt{3}-5000}{2}$$

Divide -5000+3000\sqrt{3} by 2.

$$x=1500\sqrt{3}-2500$$

Now solve the equation x=\frac{-5000±3000\sqrt{3}}{2} when ± is minus. Subtract 3000\sqrt{3} from -5000.

$$x=\frac{-3000\sqrt{3}-5000}{2}$$

Divide -5000-3000\sqrt{3} by 2.

$$x=-1500\sqrt{3}-2500$$

The equation is now solved.

$$x=1500\sqrt{3}-2500 x=-1500\sqrt{3}-2500$$

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.

$$x^{2}+5000x-500000=0$$

Add 500000 to both sides of the equation.

$$x^{2}+5000x-500000-\left(-500000\right)=-\left(-500000\right)$$

Subtracting -500000 from itself leaves 0.

$$x^{2}+5000x=-\left(-500000\right)$$

Subtract -500000 from 0.

$$x^{2}+5000x=500000$$

Divide 5000, the coefficient of the x term, by 2 to get 2500. Then add the square of 2500 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5000x+2500^{2}=500000+2500^{2}$$

Square 2500.

$$x^{2}+5000x+6250000=500000+6250000$$

Add 500000 to 6250000.

$$x^{2}+5000x+6250000=6750000$$

Factor x^{2}+5000x+6250000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2500\right)^{2}=6750000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2500\right)^{2}}=\sqrt{6750000}$$

Simplify.

$$x+2500=1500\sqrt{3} x+2500=-1500\sqrt{3}$$

Subtract 2500 from both sides of the equation.

$$x=1500\sqrt{3}-2500 x=-1500\sqrt{3}-2500$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +5000x -500000 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -5000 rs = -500000$$

Two numbers r and s sum up to -5000 exactly when the average of the two numbers is \frac{1}{2}*-5000 = -2500. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -2500 - u s = -2500 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -500000

$$(-2500 - u) (-2500 + u) = -500000$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$6250000 - u^2 = -500000$$

Simplify the expression by subtracting 6250000 on both sides

$$-u^2 = -500000-6250000 = -6750000$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 6750000 u = \pm\sqrt{6750000} = \pm \sqrt{6750000}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-2500 - \sqrt{6750000} = -5098.076 s = -2500 + \sqrt{6750000} = 98.076$$