Solve -103x^2=112-28x

$$- 103 x ^ { 2 } = 112 - 28 x$$

Solve for x (complex solution)

$$x=\frac{-18\sqrt{35}i+14}{103}\approx 0.13592233-1.03387802i$$

$$x=\frac{14+18\sqrt{35}i}{103}\approx 0.13592233+1.03387802i$$

Steps Using the Quadratic Formula

Subtract 112 from both sides.

$$-103x^{2}-112=-28x$$

Add 28x to both sides.

$$-103x^{2}-112+28x=0$$

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.

$$-103x^{2}+28x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -103 for a, 28 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-103\right)\left(-112\right)}}{2\left(-103\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-103\right)\left(-112\right)}}{2\left(-103\right)}$$

Multiply -4 times -103.

$$x=\frac{-28±\sqrt{784+412\left(-112\right)}}{2\left(-103\right)}$$

Multiply 412 times -112.

$$x=\frac{-28±\sqrt{784-46144}}{2\left(-103\right)}$$

Add 784 to -46144.

$$x=\frac{-28±\sqrt{-45360}}{2\left(-103\right)}$$

Take the square root of -45360.

$$x=\frac{-28±36\sqrt{35}i}{2\left(-103\right)}$$

Multiply 2 times -103.

$$x=\frac{-28±36\sqrt{35}i}{-206}$$

Now solve the equation x=\frac{-28±36\sqrt{35}i}{-206} when ± is plus. Add -28 to 36i\sqrt{35}.

$$x=\frac{-28+36\sqrt{35}i}{-206}$$

Divide -28+36i\sqrt{35} by -206.

$$x=\frac{-18\sqrt{35}i+14}{103}$$

Now solve the equation x=\frac{-28±36\sqrt{35}i}{-206} when ± is minus. Subtract 36i\sqrt{35} from -28.

$$x=\frac{-36\sqrt{35}i-28}{-206}$$

Divide -28-36i\sqrt{35} by -206.

$$x=\frac{14+18\sqrt{35}i}{103}$$

The equation is now solved.

$$x=\frac{-18\sqrt{35}i+14}{103} x=\frac{14+18\sqrt{35}i}{103}$$

Steps for Completing the Square

Add 28x to both sides.

$$-103x^{2}+28x=112$$

Divide both sides by -103.

$$\frac{-103x^{2}+28x}{-103}=\frac{112}{-103}$$

Dividing by -103 undoes the multiplication by -103.

$$x^{2}+\frac{28}{-103}x=\frac{112}{-103}$$

Divide 28 by -103.

$$x^{2}-\frac{28}{103}x=\frac{112}{-103}$$

Divide 112 by -103.

$$x^{2}-\frac{28}{103}x=-\frac{112}{103}$$

Divide -\frac{28}{103}, the coefficient of the x term, by 2 to get -\frac{14}{103}. Then add the square of -\frac{14}{103} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{103}x+\left(-\frac{14}{103}\right)^{2}=-\frac{112}{103}+\left(-\frac{14}{103}\right)^{2}$$

Square -\frac{14}{103} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{103}x+\frac{196}{10609}=-\frac{112}{103}+\frac{196}{10609}$$

Add -\frac{112}{103} to \frac{196}{10609} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{28}{103}x+\frac{196}{10609}=-\frac{11340}{10609}$$

Factor x^{2}-\frac{28}{103}x+\frac{196}{10609}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{14}{103}\right)^{2}=-\frac{11340}{10609}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{103}\right)^{2}}=\sqrt{-\frac{11340}{10609}}$$

Simplify.

$$x-\frac{14}{103}=\frac{18\sqrt{35}i}{103} x-\frac{14}{103}=-\frac{18\sqrt{35}i}{103}$$

Add \frac{14}{103} to both sides of the equation.

$$x=\frac{14+18\sqrt{35}i}{103} x=\frac{-18\sqrt{35}i+14}{103}$$