Solve (4i+w)w=5

$$( 4 i + w ) w = 5$$

Solve for w

$$w=1-2i$$

$$w=-1-2i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4i+w by w.

$$4iw+w^{2}=5$$

Subtract 5 from both sides.

$$4iw+w^{2}-5=0$$

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.

$$w^{2}+4iw-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4i for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 4i.

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

Multiply -4 times -5.

$$w=\frac{-4i±\sqrt{-16+20}}{2}$$

Add -16 to 20.

$$w=\frac{-4i±\sqrt{4}}{2}$$

Take the square root of 4.

$$w=\frac{-4i±2}{2}$$

Now solve the equation w=\frac{-4i±2}{2} when ± is plus. Add -4i to 2.

$$w=\frac{2-4i}{2}$$

Divide 2-4i by 2.

$$w=1-2i$$

Now solve the equation w=\frac{-4i±2}{2} when ± is minus. Subtract 2 from -4i.

$$w=\frac{-2-4i}{2}$$

Divide -2-4i by 2.

$$w=-1-2i$$

The equation is now solved.

$$w=1-2i w=-1-2i$$

Steps for Completing the Square

Use the distributive property to multiply 4i+w by w.

$$4iw+w^{2}=5$$

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.

$$w^{2}+4iw=5$$

Divide 4i, the coefficient of the x term, by 2 to get 2i. Then add the square of 2i to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+4iw+\left(2i\right)^{2}=5+\left(2i\right)^{2}$$

Square 2i.

$$w^{2}+4iw-4=5-4$$

Add 5 to -4.

$$w^{2}+4iw-4=1$$

Factor w^{2}+4iw-4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+2i\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+2i\right)^{2}}=\sqrt{1}$$

Simplify.

$$w+2i=1 w+2i=-1$$

Subtract 2i from both sides of the equation.

$$w=1-2i w=-1-2i$$