Solve x+5-2xx=40

$$x + 5 - 2 x x = 40$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{31}i+1}{4}\approx 0.25-4.175823272i$$

$$x=\frac{1+3\sqrt{31}i}{4}\approx 0.25+4.175823272i$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x+5-2x^{2}=40$$

Subtract 40 from both sides.

$$x+5-2x^{2}-40=0$$

Subtract 40 from 5 to get -35.

$$x-35-2x^{2}=0$$

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.

$$-2x^{2}+x-35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1 for b, and -35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\left(-35\right)}}{2\left(-2\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\left(-35\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8\left(-35\right)}}{2\left(-2\right)}$$

Multiply 8 times -35.

$$x=\frac{-1±\sqrt{1-280}}{2\left(-2\right)}$$

Add 1 to -280.

$$x=\frac{-1±\sqrt{-279}}{2\left(-2\right)}$$

Take the square root of -279.

$$x=\frac{-1±3\sqrt{31}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1±3\sqrt{31}i}{-4}$$

Now solve the equation x=\frac{-1±3\sqrt{31}i}{-4} when ± is plus. Add -1 to 3i\sqrt{31}.

$$x=\frac{-1+3\sqrt{31}i}{-4}$$

Divide -1+3i\sqrt{31} by -4.

$$x=\frac{-3\sqrt{31}i+1}{4}$$

Now solve the equation x=\frac{-1±3\sqrt{31}i}{-4} when ± is minus. Subtract 3i\sqrt{31} from -1.

$$x=\frac{-3\sqrt{31}i-1}{-4}$$

Divide -1-3i\sqrt{31} by -4.

$$x=\frac{1+3\sqrt{31}i}{4}$$

The equation is now solved.

$$x=\frac{-3\sqrt{31}i+1}{4} x=\frac{1+3\sqrt{31}i}{4}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x+5-2x^{2}=40$$

Subtract 5 from both sides.

$$x-2x^{2}=40-5$$

Subtract 5 from 40 to get 35.

$$x-2x^{2}=35$$

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.

$$-2x^{2}+x=35$$

Divide both sides by -2.

$$\frac{-2x^{2}+x}{-2}=\frac{35}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1}{-2}x=\frac{35}{-2}$$

Divide 1 by -2.

$$x^{2}-\frac{1}{2}x=\frac{35}{-2}$$

Divide 35 by -2.

$$x^{2}-\frac{1}{2}x=-\frac{35}{2}$$

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.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{35}{2}+\left(-\frac{1}{4}\right)^{2}$$

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{35}{2}+\frac{1}{16}$$

Add -\frac{35}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{279}{16}$$

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}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{279}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{279}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{3\sqrt{31}i}{4} x-\frac{1}{4}=-\frac{3\sqrt{31}i}{4}$$

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

$$x=\frac{1+3\sqrt{31}i}{4} x=\frac{-3\sqrt{31}i+1}{4}$$