Solve (5-2w)w=52

$$( 5 - 2 w ) w = 52$$

Solve for w

$$w=\frac{-\sqrt{391}i+5}{4}\approx 1.25-4.943429983i$$

$$w=\frac{5+\sqrt{391}i}{4}\approx 1.25+4.943429983i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5-2w by w.

$$5w-2w^{2}=52$$

Subtract 52 from both sides.

$$5w-2w^{2}-52=0$$

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.

$$-2w^{2}+5w-52=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 5 for b, and -52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-5±\sqrt{5^{2}-4\left(-2\right)\left(-52\right)}}{2\left(-2\right)}$$

Square 5.

$$w=\frac{-5±\sqrt{25-4\left(-2\right)\left(-52\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$w=\frac{-5±\sqrt{25+8\left(-52\right)}}{2\left(-2\right)}$$

Multiply 8 times -52.

$$w=\frac{-5±\sqrt{25-416}}{2\left(-2\right)}$$

Add 25 to -416.

$$w=\frac{-5±\sqrt{-391}}{2\left(-2\right)}$$

Take the square root of -391.

$$w=\frac{-5±\sqrt{391}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$w=\frac{-5±\sqrt{391}i}{-4}$$

Now solve the equation w=\frac{-5±\sqrt{391}i}{-4} when ± is plus. Add -5 to i\sqrt{391}.

$$w=\frac{-5+\sqrt{391}i}{-4}$$

Divide -5+i\sqrt{391} by -4.

$$w=\frac{-\sqrt{391}i+5}{4}$$

Now solve the equation w=\frac{-5±\sqrt{391}i}{-4} when ± is minus. Subtract i\sqrt{391} from -5.

$$w=\frac{-\sqrt{391}i-5}{-4}$$

Divide -5-i\sqrt{391} by -4.

$$w=\frac{5+\sqrt{391}i}{4}$$

The equation is now solved.

$$w=\frac{-\sqrt{391}i+5}{4} w=\frac{5+\sqrt{391}i}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 5-2w by w.

$$5w-2w^{2}=52$$

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.

$$-2w^{2}+5w=52$$

Divide both sides by -2.

$$\frac{-2w^{2}+5w}{-2}=\frac{52}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$w^{2}+\frac{5}{-2}w=\frac{52}{-2}$$

Divide 5 by -2.

$$w^{2}-\frac{5}{2}w=\frac{52}{-2}$$

Divide 52 by -2.

$$w^{2}-\frac{5}{2}w=-26$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-\frac{5}{2}w+\left(-\frac{5}{4}\right)^{2}=-26+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{5}{2}w+\frac{25}{16}=-26+\frac{25}{16}$$

Add -26 to \frac{25}{16}.

$$w^{2}-\frac{5}{2}w+\frac{25}{16}=-\frac{391}{16}$$

Factor w^{2}-\frac{5}{2}w+\frac{25}{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}.

$$\left(w-\frac{5}{4}\right)^{2}=-\frac{391}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{391}{16}}$$

Simplify.

$$w-\frac{5}{4}=\frac{\sqrt{391}i}{4} w-\frac{5}{4}=-\frac{\sqrt{391}i}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$w=\frac{5+\sqrt{391}i}{4} w=\frac{-\sqrt{391}i+5}{4}$$