Solve 1/4k^2+k-7/3=2*0

$$\frac { 1 } { 4 } k ^ { 2 } + k - \frac { 7 } { 3 } = 2 \cdot 0$$

Solve for k

$$k=\frac{2\sqrt{30}}{3}-2\approx 1.651483717$$

$$k=-\frac{2\sqrt{30}}{3}-2\approx -5.651483717$$

Steps Using the Quadratic Formula

Multiply 2 and 0 to get 0.

$$\frac{1}{4}k^{2}+k-\frac{7}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, 1 for b, and -\frac{7}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{4}\left(-\frac{7}{3}\right)}}{2\times \frac{1}{4}}$$

Square 1.

$$k=\frac{-1±\sqrt{1-4\times \frac{1}{4}\left(-\frac{7}{3}\right)}}{2\times \frac{1}{4}}$$

Multiply -4 times \frac{1}{4}.

$$k=\frac{-1±\sqrt{1-\left(-\frac{7}{3}\right)}}{2\times \frac{1}{4}}$$

Multiply -1 times -\frac{7}{3}.

$$k=\frac{-1±\sqrt{1+\frac{7}{3}}}{2\times \frac{1}{4}}$$

Add 1 to \frac{7}{3}.

$$k=\frac{-1±\sqrt{\frac{10}{3}}}{2\times \frac{1}{4}}$$

Take the square root of \frac{10}{3}.

$$k=\frac{-1±\frac{\sqrt{30}}{3}}{2\times \frac{1}{4}}$$

Multiply 2 times \frac{1}{4}.

$$k=\frac{-1±\frac{\sqrt{30}}{3}}{\frac{1}{2}}$$

Now solve the equation k=\frac{-1±\frac{\sqrt{30}}{3}}{\frac{1}{2}} when ± is plus. Add -1 to \frac{\sqrt{30}}{3}.

$$k=\frac{\frac{\sqrt{30}}{3}-1}{\frac{1}{2}}$$

Divide -1+\frac{\sqrt{30}}{3} by \frac{1}{2} by multiplying -1+\frac{\sqrt{30}}{3} by the reciprocal of \frac{1}{2}.

$$k=\frac{2\sqrt{30}}{3}-2$$

Now solve the equation k=\frac{-1±\frac{\sqrt{30}}{3}}{\frac{1}{2}} when ± is minus. Subtract \frac{\sqrt{30}}{3} from -1.

$$k=\frac{-\frac{\sqrt{30}}{3}-1}{\frac{1}{2}}$$

Divide -1-\frac{\sqrt{30}}{3} by \frac{1}{2} by multiplying -1-\frac{\sqrt{30}}{3} by the reciprocal of \frac{1}{2}.

$$k=-\frac{2\sqrt{30}}{3}-2$$

The equation is now solved.

$$k=\frac{2\sqrt{30}}{3}-2 k=-\frac{2\sqrt{30}}{3}-2$$

Steps for Completing the Square

Multiply 2 and 0 to get 0.

$$\frac{1}{4}k^{2}+k-\frac{7}{3}=0$$

Add \frac{7}{3} to both sides. Anything plus zero gives itself.

$$\frac{1}{4}k^{2}+k=\frac{7}{3}$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}k^{2}+k}{\frac{1}{4}}=\frac{\frac{7}{3}}{\frac{1}{4}}$$

Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.

$$k^{2}+\frac{1}{\frac{1}{4}}k=\frac{\frac{7}{3}}{\frac{1}{4}}$$

Divide 1 by \frac{1}{4} by multiplying 1 by the reciprocal of \frac{1}{4}.

$$k^{2}+4k=\frac{\frac{7}{3}}{\frac{1}{4}}$$

Divide \frac{7}{3} by \frac{1}{4} by multiplying \frac{7}{3} by the reciprocal of \frac{1}{4}.

$$k^{2}+4k=\frac{28}{3}$$

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.

$$k^{2}+4k+2^{2}=\frac{28}{3}+2^{2}$$

Square 2.

$$k^{2}+4k+4=\frac{28}{3}+4$$

Add \frac{28}{3} to 4.

$$k^{2}+4k+4=\frac{40}{3}$$

Factor k^{2}+4k+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}.

$$\left(k+2\right)^{2}=\frac{40}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+2\right)^{2}}=\sqrt{\frac{40}{3}}$$

Simplify.

$$k+2=\frac{2\sqrt{30}}{3} k+2=-\frac{2\sqrt{30}}{3}$$

Subtract 2 from both sides of the equation.

$$k=\frac{2\sqrt{30}}{3}-2 k=-\frac{2\sqrt{30}}{3}-2$$