Solve 2x^2-115x+1500=0

$$2 { x }^{ 2 } -115x+1500=0$$

Solve for x

$$x=20$$

$$x = \frac{75}{2} = 37\frac{1}{2} = 37.5$$

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.

$$2x^{2}-115x+1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -115 for b, and 1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-115\right)±\sqrt{\left(-115\right)^{2}-4\times 2\times 1500}}{2\times 2}$$

Square -115.

$$x=\frac{-\left(-115\right)±\sqrt{13225-4\times 2\times 1500}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-115\right)±\sqrt{13225-8\times 1500}}{2\times 2}$$

Multiply -8 times 1500.

$$x=\frac{-\left(-115\right)±\sqrt{13225-12000}}{2\times 2}$$

Add 13225 to -12000.

$$x=\frac{-\left(-115\right)±\sqrt{1225}}{2\times 2}$$

Take the square root of 1225.

$$x=\frac{-\left(-115\right)±35}{2\times 2}$$

The opposite of -115 is 115.

$$x=\frac{115±35}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{115±35}{4}$$

Now solve the equation x=\frac{115±35}{4} when ± is plus. Add 115 to 35.

$$x=\frac{150}{4}$$

Reduce the fraction \frac{150}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{75}{2}$$

Now solve the equation x=\frac{115±35}{4} when ± is minus. Subtract 35 from 115.

$$x=\frac{80}{4}$$

Divide 80 by 4.

$$x=20$$

The equation is now solved.

$$x=\frac{75}{2} x=20$$

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.

$$2x^{2}-115x+1500=0$$

Subtract 1500 from both sides of the equation.

$$2x^{2}-115x+1500-1500=-1500$$

Subtracting 1500 from itself leaves 0.

$$2x^{2}-115x=-1500$$

Divide both sides by 2.

$$\frac{2x^{2}-115x}{2}=-\frac{1500}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{115}{2}x=-\frac{1500}{2}$$

Divide -1500 by 2.

$$x^{2}-\frac{115}{2}x=-750$$

Divide -\frac{115}{2}, the coefficient of the x term, by 2 to get -\frac{115}{4}. Then add the square of -\frac{115}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{115}{2}x+\left(-\frac{115}{4}\right)^{2}=-750+\left(-\frac{115}{4}\right)^{2}$$

Square -\frac{115}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{115}{2}x+\frac{13225}{16}=-750+\frac{13225}{16}$$

Add -750 to \frac{13225}{16}.

$$x^{2}-\frac{115}{2}x+\frac{13225}{16}=\frac{1225}{16}$$

Factor x^{2}-\frac{115}{2}x+\frac{13225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{115}{4}\right)^{2}=\frac{1225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{115}{4}\right)^{2}}=\sqrt{\frac{1225}{16}}$$

Simplify.

$$x-\frac{115}{4}=\frac{35}{4} x-\frac{115}{4}=-\frac{35}{4}$$

Add \frac{115}{4} to both sides of the equation.

$$x=\frac{75}{2} x=20$$