Solve -left(5xright)^2+8x=2002

$$- { \left(5x \right) }^{ 2 } +8x = 2002$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{50034}i+4}{25}\approx 0.16-8.947312446i$$

$$x=\frac{4+\sqrt{50034}i}{25}\approx 0.16+8.947312446i$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$-5^{2}x^{2}+8x=2002$$

Calculate 5 to the power of 2 and get 25.

$$-25x^{2}+8x=2002$$

Subtract 2002 from both sides.

$$-25x^{2}+8x-2002=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -25 for a, 8 for b, and -2002 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-25\right)\left(-2002\right)}}{2\left(-25\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-25\right)\left(-2002\right)}}{2\left(-25\right)}$$

Multiply -4 times -25.

$$x=\frac{-8±\sqrt{64+100\left(-2002\right)}}{2\left(-25\right)}$$

Multiply 100 times -2002.

$$x=\frac{-8±\sqrt{64-200200}}{2\left(-25\right)}$$

Add 64 to -200200.

$$x=\frac{-8±\sqrt{-200136}}{2\left(-25\right)}$$

Take the square root of -200136.

$$x=\frac{-8±2\sqrt{50034}i}{2\left(-25\right)}$$

Multiply 2 times -25.

$$x=\frac{-8±2\sqrt{50034}i}{-50}$$

Now solve the equation x=\frac{-8±2\sqrt{50034}i}{-50} when ± is plus. Add -8 to 2i\sqrt{50034}.

$$x=\frac{-8+2\sqrt{50034}i}{-50}$$

Divide -8+2i\sqrt{50034} by -50.

$$x=\frac{-\sqrt{50034}i+4}{25}$$

Now solve the equation x=\frac{-8±2\sqrt{50034}i}{-50} when ± is minus. Subtract 2i\sqrt{50034} from -8.

$$x=\frac{-2\sqrt{50034}i-8}{-50}$$

Divide -8-2i\sqrt{50034} by -50.

$$x=\frac{4+\sqrt{50034}i}{25}$$

The equation is now solved.

$$x=\frac{-\sqrt{50034}i+4}{25} x=\frac{4+\sqrt{50034}i}{25}$$

Steps for Completing the Square

Expand \left(5x\right)^{2}.

$$-5^{2}x^{2}+8x=2002$$

Calculate 5 to the power of 2 and get 25.

$$-25x^{2}+8x=2002$$

Divide both sides by -25.

$$\frac{-25x^{2}+8x}{-25}=\frac{2002}{-25}$$

Dividing by -25 undoes the multiplication by -25.

$$x^{2}+\frac{8}{-25}x=\frac{2002}{-25}$$

Divide 8 by -25.

$$x^{2}-\frac{8}{25}x=\frac{2002}{-25}$$

Divide 2002 by -25.

$$x^{2}-\frac{8}{25}x=-\frac{2002}{25}$$

Divide -\frac{8}{25}, the coefficient of the x term, by 2 to get -\frac{4}{25}. Then add the square of -\frac{4}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{25}x+\left(-\frac{4}{25}\right)^{2}=-\frac{2002}{25}+\left(-\frac{4}{25}\right)^{2}$$

Square -\frac{4}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{25}x+\frac{16}{625}=-\frac{2002}{25}+\frac{16}{625}$$

Add -\frac{2002}{25} to \frac{16}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{25}x+\frac{16}{625}=-\frac{50034}{625}$$

Factor x^{2}-\frac{8}{25}x+\frac{16}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{25}\right)^{2}=-\frac{50034}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{25}\right)^{2}}=\sqrt{-\frac{50034}{625}}$$

Simplify.

$$x-\frac{4}{25}=\frac{\sqrt{50034}i}{25} x-\frac{4}{25}=-\frac{\sqrt{50034}i}{25}$$

Add \frac{4}{25} to both sides of the equation.

$$x=\frac{4+\sqrt{50034}i}{25} x=\frac{-\sqrt{50034}i+4}{25}$$