Solve 1/4k+9/4k=2

$$\frac { 1 } { 4 } k + \frac { 9 } { 4 k } = 2$$

Solve for k

$$k=\sqrt{7}+4\approx 6.645751311$$

$$k=4-\sqrt{7}\approx 1.354248689$$

Steps Using the Quadratic Formula

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4k, the least common multiple of 4,4k.

$$\frac{1}{4}k\times 4k+9=8k$$

Multiply \frac{1}{4} and 4 to get 1.

$$kk+9=8k$$

Multiply k and k to get k^{2}.

$$k^{2}+9=8k$$

Subtract 8k from both sides.

$$k^{2}+9-8k=0$$

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.

$$k^{2}-8k+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 9}}{2}$$

Square -8.

$$k=\frac{-\left(-8\right)±\sqrt{64-4\times 9}}{2}$$

Multiply -4 times 9.

$$k=\frac{-\left(-8\right)±\sqrt{64-36}}{2}$$

Add 64 to -36.

$$k=\frac{-\left(-8\right)±\sqrt{28}}{2}$$

Take the square root of 28.

$$k=\frac{-\left(-8\right)±2\sqrt{7}}{2}$$

The opposite of -8 is 8.

$$k=\frac{8±2\sqrt{7}}{2}$$

Now solve the equation k=\frac{8±2\sqrt{7}}{2} when ± is plus. Add 8 to 2\sqrt{7}.

$$k=\frac{2\sqrt{7}+8}{2}$$

Divide 8+2\sqrt{7} by 2.

$$k=\sqrt{7}+4$$

Now solve the equation k=\frac{8±2\sqrt{7}}{2} when ± is minus. Subtract 2\sqrt{7} from 8.

$$k=\frac{8-2\sqrt{7}}{2}$$

Divide 8-2\sqrt{7} by 2.

$$k=4-\sqrt{7}$$

The equation is now solved.

$$k=\sqrt{7}+4 k=4-\sqrt{7}$$

Steps for Completing the Square

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4k, the least common multiple of 4,4k.

$$\frac{1}{4}k\times 4k+9=8k$$

Multiply \frac{1}{4} and 4 to get 1.

$$kk+9=8k$$

Multiply k and k to get k^{2}.

$$k^{2}+9=8k$$

Subtract 8k from both sides.

$$k^{2}+9-8k=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$k^{2}-8k=-9$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-8k+\left(-4\right)^{2}=-9+\left(-4\right)^{2}$$

Square -4.

$$k^{2}-8k+16=-9+16$$

Add -9 to 16.

$$k^{2}-8k+16=7$$

Factor k^{2}-8k+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-4\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-4\right)^{2}}=\sqrt{7}$$

Simplify.

$$k-4=\sqrt{7} k-4=-\sqrt{7}$$

Add 4 to both sides of the equation.

$$k=\sqrt{7}+4 k=4-\sqrt{7}$$