Solve 2222x^2+12x-23=0

$$2222 { x }^{ 2 } +12x-23=0$$

Solve for x

$$x=\frac{\sqrt{51142}}{2222}-\frac{3}{1111}\approx 0.099075594$$

$$x=-\frac{\sqrt{51142}}{2222}-\frac{3}{1111}\approx -0.104476134$$

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.

$$2222x^{2}+12x-23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2222 for a, 12 for b, and -23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2222\left(-23\right)}}{2\times 2222}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2222\left(-23\right)}}{2\times 2222}$$

Multiply -4 times 2222.

$$x=\frac{-12±\sqrt{144-8888\left(-23\right)}}{2\times 2222}$$

Multiply -8888 times -23.

$$x=\frac{-12±\sqrt{144+204424}}{2\times 2222}$$

Add 144 to 204424.

$$x=\frac{-12±\sqrt{204568}}{2\times 2222}$$

Take the square root of 204568.

$$x=\frac{-12±2\sqrt{51142}}{2\times 2222}$$

Multiply 2 times 2222.

$$x=\frac{-12±2\sqrt{51142}}{4444}$$

Now solve the equation x=\frac{-12±2\sqrt{51142}}{4444} when ± is plus. Add -12 to 2\sqrt{51142}.

$$x=\frac{2\sqrt{51142}-12}{4444}$$

Divide -12+2\sqrt{51142} by 4444.

$$x=\frac{\sqrt{51142}}{2222}-\frac{3}{1111}$$

Now solve the equation x=\frac{-12±2\sqrt{51142}}{4444} when ± is minus. Subtract 2\sqrt{51142} from -12.

$$x=\frac{-2\sqrt{51142}-12}{4444}$$

Divide -12-2\sqrt{51142} by 4444.

$$x=-\frac{\sqrt{51142}}{2222}-\frac{3}{1111}$$

The equation is now solved.

$$x=\frac{\sqrt{51142}}{2222}-\frac{3}{1111} x=-\frac{\sqrt{51142}}{2222}-\frac{3}{1111}$$

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.

$$2222x^{2}+12x-23=0$$

Add 23 to both sides of the equation.

$$2222x^{2}+12x-23-\left(-23\right)=-\left(-23\right)$$

Subtracting -23 from itself leaves 0.

$$2222x^{2}+12x=-\left(-23\right)$$

Subtract -23 from 0.

$$2222x^{2}+12x=23$$

Divide both sides by 2222.

$$\frac{2222x^{2}+12x}{2222}=\frac{23}{2222}$$

Dividing by 2222 undoes the multiplication by 2222.

$$x^{2}+\frac{12}{2222}x=\frac{23}{2222}$$

Reduce the fraction \frac{12}{2222} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{6}{1111}x=\frac{23}{2222}$$

Divide \frac{6}{1111}, the coefficient of the x term, by 2 to get \frac{3}{1111}. Then add the square of \frac{3}{1111} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6}{1111}x+\left(\frac{3}{1111}\right)^{2}=\frac{23}{2222}+\left(\frac{3}{1111}\right)^{2}$$

Square \frac{3}{1111} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{1111}x+\frac{9}{1234321}=\frac{23}{2222}+\frac{9}{1234321}$$

Add \frac{23}{2222} to \frac{9}{1234321} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{1111}x+\frac{9}{1234321}=\frac{25571}{2468642}$$

Factor x^{2}+\frac{6}{1111}x+\frac{9}{1234321}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{1111}\right)^{2}=\frac{25571}{2468642}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{1111}\right)^{2}}=\sqrt{\frac{25571}{2468642}}$$

Simplify.

$$x+\frac{3}{1111}=\frac{\sqrt{51142}}{2222} x+\frac{3}{1111}=-\frac{\sqrt{51142}}{2222}$$

Subtract \frac{3}{1111} from both sides of the equation.

$$x=\frac{\sqrt{51142}}{2222}-\frac{3}{1111} x=-\frac{\sqrt{51142}}{2222}-\frac{3}{1111}$$