Solve 4+w^2+10-6w+w^2=18

$$4 + w ^ { 2 } + 10 - 6 w + w ^ { 2 } = 18$$

Solve for w

$$w = \frac{\sqrt{17} + 3}{2} \approx 3.561552813$$

$$w=\frac{3-\sqrt{17}}{2}\approx -0.561552813$$

Steps Using the Quadratic Formula

Add 4 and 10 to get 14.

$$14+w^{2}-6w+w^{2}=18$$

Combine w^{2} and w^{2} to get 2w^{2}.

$$14+2w^{2}-6w=18$$

Subtract 18 from both sides.

$$14+2w^{2}-6w-18=0$$

Subtract 18 from 14 to get -4.

$$-4+2w^{2}-6w=0$$

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.

$$2w^{2}-6w-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -6 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-4\right)}}{2\times 2}$$

Square -6.

$$w=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-4\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-\left(-6\right)±\sqrt{36-8\left(-4\right)}}{2\times 2}$$

Multiply -8 times -4.

$$w=\frac{-\left(-6\right)±\sqrt{36+32}}{2\times 2}$$

Add 36 to 32.

$$w=\frac{-\left(-6\right)±\sqrt{68}}{2\times 2}$$

Take the square root of 68.

$$w=\frac{-\left(-6\right)±2\sqrt{17}}{2\times 2}$$

The opposite of -6 is 6.

$$w=\frac{6±2\sqrt{17}}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{6±2\sqrt{17}}{4}$$

Now solve the equation w=\frac{6±2\sqrt{17}}{4} when ± is plus. Add 6 to 2\sqrt{17}.

$$w=\frac{2\sqrt{17}+6}{4}$$

Divide 6+2\sqrt{17} by 4.

$$w=\frac{\sqrt{17}+3}{2}$$

Now solve the equation w=\frac{6±2\sqrt{17}}{4} when ± is minus. Subtract 2\sqrt{17} from 6.

$$w=\frac{6-2\sqrt{17}}{4}$$

Divide 6-2\sqrt{17} by 4.

$$w=\frac{3-\sqrt{17}}{2}$$

The equation is now solved.

$$w=\frac{\sqrt{17}+3}{2} w=\frac{3-\sqrt{17}}{2}$$

Steps for Completing the Square

Add 4 and 10 to get 14.

$$14+w^{2}-6w+w^{2}=18$$

Combine w^{2} and w^{2} to get 2w^{2}.

$$14+2w^{2}-6w=18$$

Subtract 14 from both sides.

$$2w^{2}-6w=18-14$$

Subtract 14 from 18 to get 4.

$$2w^{2}-6w=4$$

Divide both sides by 2.

$$\frac{2w^{2}-6w}{2}=\frac{4}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$w^{2}+\left(-\frac{6}{2}\right)w=\frac{4}{2}$$

Divide -6 by 2.

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

Divide 4 by 2.

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

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-3w+\frac{9}{4}=2+\frac{9}{4}$$

Add 2 to \frac{9}{4}.

$$w^{2}-3w+\frac{9}{4}=\frac{17}{4}$$

Factor w^{2}-3w+\frac{9}{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}.

$$\left(w-\frac{3}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{3}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$w-\frac{3}{2}=\frac{\sqrt{17}}{2} w-\frac{3}{2}=-\frac{\sqrt{17}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$w=\frac{\sqrt{17}+3}{2} w=\frac{3-\sqrt{17}}{2}$$