Solve x^2-5x^2+2x-56=5088

$$x ^ { 2 } - 5 x ^ { 2 } + 2 x - 56 = 5088$$

Solve for x (complex solution)

$$x=\frac{1+5\sqrt{823}i}{4}\approx 0.25+35.859970719i$$

$$x=\frac{-5\sqrt{823}i+1}{4}\approx 0.25-35.859970719i$$

Steps Using the Quadratic Formula

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+2x-56=5088$$

Subtract 5088 from both sides.

$$-4x^{2}+2x-56-5088=0$$

Subtract 5088 from -56 to get -5144.

$$-4x^{2}+2x-5144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 2 for b, and -5144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-4\right)\left(-5144\right)}}{2\left(-4\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-4\right)\left(-5144\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-2±\sqrt{4+16\left(-5144\right)}}{2\left(-4\right)}$$

Multiply 16 times -5144.

$$x=\frac{-2±\sqrt{4-82304}}{2\left(-4\right)}$$

Add 4 to -82304.

$$x=\frac{-2±\sqrt{-82300}}{2\left(-4\right)}$$

Take the square root of -82300.

$$x=\frac{-2±10\sqrt{823}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-2±10\sqrt{823}i}{-8}$$

Now solve the equation x=\frac{-2±10\sqrt{823}i}{-8} when ± is plus. Add -2 to 10i\sqrt{823}.

$$x=\frac{-2+10\sqrt{823}i}{-8}$$

Divide -2+10i\sqrt{823} by -8.

$$x=\frac{-5\sqrt{823}i+1}{4}$$

Now solve the equation x=\frac{-2±10\sqrt{823}i}{-8} when ± is minus. Subtract 10i\sqrt{823} from -2.

$$x=\frac{-10\sqrt{823}i-2}{-8}$$

Divide -2-10i\sqrt{823} by -8.

$$x=\frac{1+5\sqrt{823}i}{4}$$

The equation is now solved.

$$x=\frac{-5\sqrt{823}i+1}{4} x=\frac{1+5\sqrt{823}i}{4}$$

Steps for Completing the Square

Combine x^{2} and -5x^{2} to get -4x^{2}.

$$-4x^{2}+2x-56=5088$$

Add 56 to both sides.

$$-4x^{2}+2x=5088+56$$

Add 5088 and 56 to get 5144.

$$-4x^{2}+2x=5144$$

Divide both sides by -4.

$$\frac{-4x^{2}+2x}{-4}=\frac{5144}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{2}{-4}x=\frac{5144}{-4}$$

Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{2}x=\frac{5144}{-4}$$

Divide 5144 by -4.

$$x^{2}-\frac{1}{2}x=-1286$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-1286+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-1286+\frac{1}{16}$$

Add -1286 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{20575}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{20575}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{20575}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{5\sqrt{823}i}{4} x-\frac{1}{4}=-\frac{5\sqrt{823}i}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1+5\sqrt{823}i}{4} x=\frac{-5\sqrt{823}i+1}{4}$$