Solve 2x^2-298x+6000=0

$$2 { x }^{ 2 } -298x+6000 = 0$$

Solve for x

$$x=24$$

$$x=125$$

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.

$$2x^{2}-298x+6000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -298 for b, and 6000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-298\right)±\sqrt{\left(-298\right)^{2}-4\times 2\times 6000}}{2\times 2}$$

Square -298.

$$x=\frac{-\left(-298\right)±\sqrt{88804-4\times 2\times 6000}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-298\right)±\sqrt{88804-8\times 6000}}{2\times 2}$$

Multiply -8 times 6000.

$$x=\frac{-\left(-298\right)±\sqrt{88804-48000}}{2\times 2}$$

Add 88804 to -48000.

$$x=\frac{-\left(-298\right)±\sqrt{40804}}{2\times 2}$$

Take the square root of 40804.

$$x=\frac{-\left(-298\right)±202}{2\times 2}$$

The opposite of -298 is 298.

$$x=\frac{298±202}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{298±202}{4}$$

Now solve the equation x=\frac{298±202}{4} when ± is plus. Add 298 to 202.

$$x=\frac{500}{4}$$

Divide 500 by 4.

$$x=125$$

Now solve the equation x=\frac{298±202}{4} when ± is minus. Subtract 202 from 298.

$$x=\frac{96}{4}$$

Divide 96 by 4.

$$x=24$$

The equation is now solved.

$$x=125 x=24$$

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.

$$2x^{2}-298x+6000=0$$

Subtract 6000 from both sides of the equation.

$$2x^{2}-298x+6000-6000=-6000$$

Subtracting 6000 from itself leaves 0.

$$2x^{2}-298x=-6000$$

Divide both sides by 2.

$$\frac{2x^{2}-298x}{2}=-\frac{6000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{298}{2}\right)x=-\frac{6000}{2}$$

Divide -298 by 2.

$$x^{2}-149x=-\frac{6000}{2}$$

Divide -6000 by 2.

$$x^{2}-149x=-3000$$

Divide -149, the coefficient of the x term, by 2 to get -\frac{149}{2}. Then add the square of -\frac{149}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-149x+\left(-\frac{149}{2}\right)^{2}=-3000+\left(-\frac{149}{2}\right)^{2}$$

Square -\frac{149}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-149x+\frac{22201}{4}=-3000+\frac{22201}{4}$$

Add -3000 to \frac{22201}{4}.

$$x^{2}-149x+\frac{22201}{4}=\frac{10201}{4}$$

Factor x^{2}-149x+\frac{22201}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{149}{2}\right)^{2}=\frac{10201}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{149}{2}\right)^{2}}=\sqrt{\frac{10201}{4}}$$

Simplify.

$$x-\frac{149}{2}=\frac{101}{2} x-\frac{149}{2}=-\frac{101}{2}$$

Add \frac{149}{2} to both sides of the equation.

$$x=125 x=24$$