Solve 2x^2-5000x+9000=0

$$2 x ^ { 2 } - 5000 x + 9000 = 0$$

Solve for x

$$x=20\sqrt{3895}+1250\approx 2498.19870213$$

$$x=1250-20\sqrt{3895}\approx 1.80129787$$

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.

$$2x^{2}-5000x+9000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5000 for b, and 9000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5000\right)±\sqrt{\left(-5000\right)^{2}-4\times 2\times 9000}}{2\times 2}$$

Square -5000.

$$x=\frac{-\left(-5000\right)±\sqrt{25000000-4\times 2\times 9000}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-5000\right)±\sqrt{25000000-8\times 9000}}{2\times 2}$$

Multiply -8 times 9000.

$$x=\frac{-\left(-5000\right)±\sqrt{25000000-72000}}{2\times 2}$$

Add 25000000 to -72000.

$$x=\frac{-\left(-5000\right)±\sqrt{24928000}}{2\times 2}$$

Take the square root of 24928000.

$$x=\frac{-\left(-5000\right)±80\sqrt{3895}}{2\times 2}$$

The opposite of -5000 is 5000.

$$x=\frac{5000±80\sqrt{3895}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{5000±80\sqrt{3895}}{4}$$

Now solve the equation x=\frac{5000±80\sqrt{3895}}{4} when ± is plus. Add 5000 to 80\sqrt{3895}.

$$x=\frac{80\sqrt{3895}+5000}{4}$$

Divide 5000+80\sqrt{3895} by 4.

$$x=20\sqrt{3895}+1250$$

Now solve the equation x=\frac{5000±80\sqrt{3895}}{4} when ± is minus. Subtract 80\sqrt{3895} from 5000.

$$x=\frac{5000-80\sqrt{3895}}{4}$$

Divide 5000-80\sqrt{3895} by 4.

$$x=1250-20\sqrt{3895}$$

The equation is now solved.

$$x=20\sqrt{3895}+1250 x=1250-20\sqrt{3895}$$

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.

$$2x^{2}-5000x+9000=0$$

Subtract 9000 from both sides of the equation.

$$2x^{2}-5000x+9000-9000=-9000$$

Subtracting 9000 from itself leaves 0.

$$2x^{2}-5000x=-9000$$

Divide both sides by 2.

$$\frac{2x^{2}-5000x}{2}=-\frac{9000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{5000}{2}\right)x=-\frac{9000}{2}$$

Divide -5000 by 2.

$$x^{2}-2500x=-\frac{9000}{2}$$

Divide -9000 by 2.

$$x^{2}-2500x=-4500$$

Divide -2500, the coefficient of the x term, by 2 to get -1250. Then add the square of -1250 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2500x+\left(-1250\right)^{2}=-4500+\left(-1250\right)^{2}$$

Square -1250.

$$x^{2}-2500x+1562500=-4500+1562500$$

Add -4500 to 1562500.

$$x^{2}-2500x+1562500=1558000$$

Factor x^{2}-2500x+1562500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1250\right)^{2}=1558000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1250\right)^{2}}=\sqrt{1558000}$$

Simplify.

$$x-1250=20\sqrt{3895} x-1250=-20\sqrt{3895}$$

Add 1250 to both sides of the equation.

$$x=20\sqrt{3895}+1250 x=1250-20\sqrt{3895}$$

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.This is achieved by dividing both sides of the equation by 2

$$x ^ 2 -2500x +4500 = 0$$

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

$$r + s = 2500 rs = 4500$$

Two numbers r and s sum up to 2500 exactly when the average of the two numbers is \frac{1}{2}*2500 = 1250. 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>

$$r = 1250 - u s = 1250 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 4500

$$(1250 - u) (1250 + u) = 4500$$

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

$$1562500 - u^2 = 4500$$

Simplify the expression by subtracting 1562500 on both sides

$$-u^2 = 4500-1562500 = -1558000$$

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

$$u^2 = 1558000 u = \pm\sqrt{1558000} = \pm \sqrt{1558000}$$

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

$$r =1250 - \sqrt{1558000} = 1.801 s = 1250 + \sqrt{1558000} = 2498.199$$