Solve x^2-379x-188=303

$$x ^ { 2 } - 379 x - 188 = 303$$

Solve for x

$$x = \frac{\sqrt{145605} + 379}{2} \approx 380.291116145$$

$$x=\frac{379-\sqrt{145605}}{2}\approx -1.291116145$$

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.

$$x^{2}-379x-188=303$$

Subtract 303 from both sides of the equation.

$$x^{2}-379x-188-303=303-303$$

Subtracting 303 from itself leaves 0.

$$x^{2}-379x-188-303=0$$

Subtract 303 from -188.

$$x^{2}-379x-491=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -379 for b, and -491 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-379\right)±\sqrt{\left(-379\right)^{2}-4\left(-491\right)}}{2}$$

Square -379.

$$x=\frac{-\left(-379\right)±\sqrt{143641-4\left(-491\right)}}{2}$$

Multiply -4 times -491.

$$x=\frac{-\left(-379\right)±\sqrt{143641+1964}}{2}$$

Add 143641 to 1964.

$$x=\frac{-\left(-379\right)±\sqrt{145605}}{2}$$

The opposite of -379 is 379.

$$x=\frac{379±\sqrt{145605}}{2}$$

Now solve the equation x=\frac{379±\sqrt{145605}}{2} when ± is plus. Add 379 to \sqrt{145605}.

$$x=\frac{\sqrt{145605}+379}{2}$$

Now solve the equation x=\frac{379±\sqrt{145605}}{2} when ± is minus. Subtract \sqrt{145605} from 379.

$$x=\frac{379-\sqrt{145605}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{145605}+379}{2} x=\frac{379-\sqrt{145605}}{2}$$

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.

$$x^{2}-379x-188=303$$

Add 188 to both sides of the equation.

$$x^{2}-379x-188-\left(-188\right)=303-\left(-188\right)$$

Subtracting -188 from itself leaves 0.

$$x^{2}-379x=303-\left(-188\right)$$

Subtract -188 from 303.

$$x^{2}-379x=491$$

Divide -379, the coefficient of the x term, by 2 to get -\frac{379}{2}. Then add the square of -\frac{379}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-379x+\left(-\frac{379}{2}\right)^{2}=491+\left(-\frac{379}{2}\right)^{2}$$

Square -\frac{379}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-379x+\frac{143641}{4}=491+\frac{143641}{4}$$

Add 491 to \frac{143641}{4}.

$$x^{2}-379x+\frac{143641}{4}=\frac{145605}{4}$$

Factor x^{2}-379x+\frac{143641}{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}.

$$\left(x-\frac{379}{2}\right)^{2}=\frac{145605}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{379}{2}\right)^{2}}=\sqrt{\frac{145605}{4}}$$

Simplify.

$$x-\frac{379}{2}=\frac{\sqrt{145605}}{2} x-\frac{379}{2}=-\frac{\sqrt{145605}}{2}$$

Add \frac{379}{2} to both sides of the equation.

$$x=\frac{\sqrt{145605}+379}{2} x=\frac{379-\sqrt{145605}}{2}$$