Solve 2*2017=2k^2+6k

$$2 \times 2017 = 2 k ^ { 2 } + 6 k$$

Solve for k

$$k = \frac{\sqrt{8077} - 3}{2} \approx 43.436065693$$

$$k=\frac{-\sqrt{8077}-3}{2}\approx -46.436065693$$

Steps Using the Quadratic Formula

Multiply 2 and 2017 to get 4034.

$$4034=2k^{2}+6k$$

Swap sides so that all variable terms are on the left hand side.

$$2k^{2}+6k=4034$$

Subtract 4034 from both sides.

$$2k^{2}+6k-4034=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 6 for b, and -4034 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-6±\sqrt{6^{2}-4\times 2\left(-4034\right)}}{2\times 2}$$

Square 6.

$$k=\frac{-6±\sqrt{36-4\times 2\left(-4034\right)}}{2\times 2}$$

Multiply -4 times 2.

$$k=\frac{-6±\sqrt{36-8\left(-4034\right)}}{2\times 2}$$

Multiply -8 times -4034.

$$k=\frac{-6±\sqrt{36+32272}}{2\times 2}$$

Add 36 to 32272.

$$k=\frac{-6±\sqrt{32308}}{2\times 2}$$

Take the square root of 32308.

$$k=\frac{-6±2\sqrt{8077}}{2\times 2}$$

Multiply 2 times 2.

$$k=\frac{-6±2\sqrt{8077}}{4}$$

Now solve the equation k=\frac{-6±2\sqrt{8077}}{4} when ± is plus. Add -6 to 2\sqrt{8077}.

$$k=\frac{2\sqrt{8077}-6}{4}$$

Divide -6+2\sqrt{8077} by 4.

$$k=\frac{\sqrt{8077}-3}{2}$$

Now solve the equation k=\frac{-6±2\sqrt{8077}}{4} when ± is minus. Subtract 2\sqrt{8077} from -6.

$$k=\frac{-2\sqrt{8077}-6}{4}$$

Divide -6-2\sqrt{8077} by 4.

$$k=\frac{-\sqrt{8077}-3}{2}$$

The equation is now solved.

$$k=\frac{\sqrt{8077}-3}{2} k=\frac{-\sqrt{8077}-3}{2}$$

Steps for Completing the Square

Multiply 2 and 2017 to get 4034.

$$4034=2k^{2}+6k$$

Swap sides so that all variable terms are on the left hand side.

$$2k^{2}+6k=4034$$

Divide both sides by 2.

$$\frac{2k^{2}+6k}{2}=\frac{4034}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$k^{2}+\frac{6}{2}k=\frac{4034}{2}$$

Divide 6 by 2.

$$k^{2}+3k=\frac{4034}{2}$$

Divide 4034 by 2.

$$k^{2}+3k=2017$$

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.

$$k^{2}+3k+\left(\frac{3}{2}\right)^{2}=2017+\left(\frac{3}{2}\right)^{2}$$

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

$$k^{2}+3k+\frac{9}{4}=2017+\frac{9}{4}$$

Add 2017 to \frac{9}{4}.

$$k^{2}+3k+\frac{9}{4}=\frac{8077}{4}$$

Factor k^{2}+3k+\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}.

$$\left(k+\frac{3}{2}\right)^{2}=\frac{8077}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+\frac{3}{2}\right)^{2}}=\sqrt{\frac{8077}{4}}$$

Simplify.

$$k+\frac{3}{2}=\frac{\sqrt{8077}}{2} k+\frac{3}{2}=-\frac{\sqrt{8077}}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$k=\frac{\sqrt{8077}-3}{2} k=\frac{-\sqrt{8077}-3}{2}$$