Solve x^2-3x+214=60600

$$x ^ { 2 } - 3 x + 214 = 60600$$

Solve for x

$$x = \frac{\sqrt{241553} + 3}{2} \approx 247.24020835$$

$$x=\frac{3-\sqrt{241553}}{2}\approx -244.24020835$$

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.

$$x^{2}-3x+214=60600$$

Subtract 60600 from both sides of the equation.

$$x^{2}-3x+214-60600=60600-60600$$

Subtracting 60600 from itself leaves 0.

$$x^{2}-3x+214-60600=0$$

Subtract 60600 from 214.

$$x^{2}-3x-60386=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -60386 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-60386\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-60386\right)}}{2}$$

Multiply -4 times -60386.

$$x=\frac{-\left(-3\right)±\sqrt{9+241544}}{2}$$

Add 9 to 241544.

$$x=\frac{-\left(-3\right)±\sqrt{241553}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{241553}}{2}$$

Now solve the equation x=\frac{3±\sqrt{241553}}{2} when ± is plus. Add 3 to \sqrt{241553}.

$$x=\frac{\sqrt{241553}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{241553}}{2} when ± is minus. Subtract \sqrt{241553} from 3.

$$x=\frac{3-\sqrt{241553}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{241553}+3}{2} x=\frac{3-\sqrt{241553}}{2}$$

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.

$$x^{2}-3x+214=60600$$

Subtract 214 from both sides of the equation.

$$x^{2}-3x+214-214=60600-214$$

Subtracting 214 from itself leaves 0.

$$x^{2}-3x=60600-214$$

Subtract 214 from 60600.

$$x^{2}-3x=60386$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=60386+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=60386+\frac{9}{4}$$

Add 60386 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{241553}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{241553}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{241553}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{241553}}{2} x-\frac{3}{2}=-\frac{\sqrt{241553}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{\sqrt{241553}+3}{2} x=\frac{3-\sqrt{241553}}{2}$$