Solve x^2-350x+25000=0

$${ x }^{ 2 } -350x+25000=0$$

Solve for x

$$x=100$$

$$x=250$$

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.

$$x^{2}-350x+25000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -350 for b, and 25000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-350\right)±\sqrt{\left(-350\right)^{2}-4\times 25000}}{2}$$

Square -350.

$$x=\frac{-\left(-350\right)±\sqrt{122500-4\times 25000}}{2}$$

Multiply -4 times 25000.

$$x=\frac{-\left(-350\right)±\sqrt{122500-100000}}{2}$$

Add 122500 to -100000.

$$x=\frac{-\left(-350\right)±\sqrt{22500}}{2}$$

Take the square root of 22500.

$$x=\frac{-\left(-350\right)±150}{2}$$

The opposite of -350 is 350.

$$x=\frac{350±150}{2}$$

Now solve the equation x=\frac{350±150}{2} when ± is plus. Add 350 to 150.

$$x=\frac{500}{2}$$

Divide 500 by 2.

$$x=250$$

Now solve the equation x=\frac{350±150}{2} when ± is minus. Subtract 150 from 350.

$$x=\frac{200}{2}$$

Divide 200 by 2.

$$x=100$$

The equation is now solved.

$$x=250 x=100$$

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.

$$x^{2}-350x+25000=0$$

Subtract 25000 from both sides of the equation.

$$x^{2}-350x+25000-25000=-25000$$

Subtracting 25000 from itself leaves 0.

$$x^{2}-350x=-25000$$

Divide -350, the coefficient of the x term, by 2 to get -175. Then add the square of -175 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-350x+\left(-175\right)^{2}=-25000+\left(-175\right)^{2}$$

Square -175.

$$x^{2}-350x+30625=-25000+30625$$

Add -25000 to 30625.

$$x^{2}-350x+30625=5625$$

Factor x^{2}-350x+30625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-175\right)^{2}=5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-175\right)^{2}}=\sqrt{5625}$$

Simplify.

$$x-175=75 x-175=-75$$

Add 175 to both sides of the equation.

$$x=250 x=100$$