Solve 992x^2+42x+125=43

$$992 x ^ { 2 } + 42 x + 125 = 43$$

Solve for x (complex solution)

$$x=\frac{-21+\sqrt{80903}i}{992}\approx -0.021169355+0.286728354i$$

$$x=\frac{-\sqrt{80903}i-21}{992}\approx -0.021169355-0.286728354i$$

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.

$$992x^{2}+42x+125=43$$

Subtract 43 from both sides of the equation.

$$992x^{2}+42x+125-43=43-43$$

Subtracting 43 from itself leaves 0.

$$992x^{2}+42x+125-43=0$$

Subtract 43 from 125.

$$992x^{2}+42x+82=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 992 for a, 42 for b, and 82 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-42±\sqrt{42^{2}-4\times 992\times 82}}{2\times 992}$$

Square 42.

$$x=\frac{-42±\sqrt{1764-4\times 992\times 82}}{2\times 992}$$

Multiply -4 times 992.

$$x=\frac{-42±\sqrt{1764-3968\times 82}}{2\times 992}$$

Multiply -3968 times 82.

$$x=\frac{-42±\sqrt{1764-325376}}{2\times 992}$$

Add 1764 to -325376.

$$x=\frac{-42±\sqrt{-323612}}{2\times 992}$$

Take the square root of -323612.

$$x=\frac{-42±2\sqrt{80903}i}{2\times 992}$$

Multiply 2 times 992.

$$x=\frac{-42±2\sqrt{80903}i}{1984}$$

Now solve the equation x=\frac{-42±2\sqrt{80903}i}{1984} when ± is plus. Add -42 to 2i\sqrt{80903}.

$$x=\frac{-42+2\sqrt{80903}i}{1984}$$

Divide -42+2i\sqrt{80903} by 1984.

$$x=\frac{-21+\sqrt{80903}i}{992}$$

Now solve the equation x=\frac{-42±2\sqrt{80903}i}{1984} when ± is minus. Subtract 2i\sqrt{80903} from -42.

$$x=\frac{-2\sqrt{80903}i-42}{1984}$$

Divide -42-2i\sqrt{80903} by 1984.

$$x=\frac{-\sqrt{80903}i-21}{992}$$

The equation is now solved.

$$x=\frac{-21+\sqrt{80903}i}{992} x=\frac{-\sqrt{80903}i-21}{992}$$

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.

$$992x^{2}+42x+125=43$$

Subtract 125 from both sides of the equation.

$$992x^{2}+42x+125-125=43-125$$

Subtracting 125 from itself leaves 0.

$$992x^{2}+42x=43-125$$

Subtract 125 from 43.

$$992x^{2}+42x=-82$$

Divide both sides by 992.

$$\frac{992x^{2}+42x}{992}=-\frac{82}{992}$$

Dividing by 992 undoes the multiplication by 992.

$$x^{2}+\frac{42}{992}x=-\frac{82}{992}$$

Reduce the fraction \frac{42}{992} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{21}{496}x=-\frac{82}{992}$$

Reduce the fraction \frac{-82}{992} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{21}{496}x=-\frac{41}{496}$$

Divide \frac{21}{496}, the coefficient of the x term, by 2 to get \frac{21}{992}. Then add the square of \frac{21}{992} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{21}{496}x+\left(\frac{21}{992}\right)^{2}=-\frac{41}{496}+\left(\frac{21}{992}\right)^{2}$$

Square \frac{21}{992} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{21}{496}x+\frac{441}{984064}=-\frac{41}{496}+\frac{441}{984064}$$

Add -\frac{41}{496} to \frac{441}{984064} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{21}{496}x+\frac{441}{984064}=-\frac{80903}{984064}$$

Factor x^{2}+\frac{21}{496}x+\frac{441}{984064}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{21}{992}\right)^{2}=-\frac{80903}{984064}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{21}{992}\right)^{2}}=\sqrt{-\frac{80903}{984064}}$$

Simplify.

$$x+\frac{21}{992}=\frac{\sqrt{80903}i}{992} x+\frac{21}{992}=-\frac{\sqrt{80903}i}{992}$$

Subtract \frac{21}{992} from both sides of the equation.

$$x=\frac{-21+\sqrt{80903}i}{992} x=\frac{-\sqrt{80903}i-21}{992}$$