Solve w^2=4w-1

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

Solve for w

$$w=\sqrt{3}+2\approx 3.732050808$$

$$w=2-\sqrt{3}\approx 0.267949192$$

Steps Using the Quadratic Formula

Subtract 4w from both sides.

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

Add 1 to both sides.

$$w^{2}-4w+1=0$$

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

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

Square -4.

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

Add 16 to -4.

$$w=\frac{-\left(-4\right)±\sqrt{12}}{2}$$

Take the square root of 12.

$$w=\frac{-\left(-4\right)±2\sqrt{3}}{2}$$

The opposite of -4 is 4.

$$w=\frac{4±2\sqrt{3}}{2}$$

Now solve the equation w=\frac{4±2\sqrt{3}}{2} when ± is plus. Add 4 to 2\sqrt{3}.

$$w=\frac{2\sqrt{3}+4}{2}$$

Divide 4+2\sqrt{3} by 2.

$$w=\sqrt{3}+2$$

Now solve the equation w=\frac{4±2\sqrt{3}}{2} when ± is minus. Subtract 2\sqrt{3} from 4.

$$w=\frac{4-2\sqrt{3}}{2}$$

Divide 4-2\sqrt{3} by 2.

$$w=2-\sqrt{3}$$

The equation is now solved.

$$w=\sqrt{3}+2 w=2-\sqrt{3}$$

Steps for Completing the Square

Subtract 4w from both sides.

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

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

$$w^{2}-4w+\left(-2\right)^{2}=-1+\left(-2\right)^{2}$$

Square -2.

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

Add -1 to 4.

$$w^{2}-4w+4=3$$

Factor w^{2}-4w+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}.

$$\left(w-2\right)^{2}=3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-2\right)^{2}}=\sqrt{3}$$

Simplify.

$$w-2=\sqrt{3} w-2=-\sqrt{3}$$

Add 2 to both sides of the equation.

$$w=\sqrt{3}+2 w=2-\sqrt{3}$$