Solve 1/4k+9/4k=3

$$\frac { 1 } { 4 } k + \frac { 9 } { 4 k } = 3$$

Solve for k

$$k=3\sqrt{3}+6\approx 11.196152423$$

$$k=6-3\sqrt{3}\approx 0.803847577$$

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.

$$\frac{1}{4}k\times 4k+9=12k$$

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

$$kk+9=12k$$

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

$$k^{2}+9=12k$$

Subtract 12k from both sides.

$$k^{2}+9-12k=0$$

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.

$$k^{2}-12k+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 9}}{2}$$

Square -12.

$$k=\frac{-\left(-12\right)±\sqrt{144-4\times 9}}{2}$$

Multiply -4 times 9.

$$k=\frac{-\left(-12\right)±\sqrt{144-36}}{2}$$

Add 144 to -36.

$$k=\frac{-\left(-12\right)±\sqrt{108}}{2}$$

Take the square root of 108.

$$k=\frac{-\left(-12\right)±6\sqrt{3}}{2}$$

The opposite of -12 is 12.

$$k=\frac{12±6\sqrt{3}}{2}$$

Now solve the equation k=\frac{12±6\sqrt{3}}{2} when ± is plus. Add 12 to 6\sqrt{3}.

$$k=\frac{6\sqrt{3}+12}{2}$$

Divide 12+6\sqrt{3} by 2.

$$k=3\sqrt{3}+6$$

Now solve the equation k=\frac{12±6\sqrt{3}}{2} when ± is minus. Subtract 6\sqrt{3} from 12.

$$k=\frac{12-6\sqrt{3}}{2}$$

Divide 12-6\sqrt{3} by 2.

$$k=6-3\sqrt{3}$$

The equation is now solved.

$$k=3\sqrt{3}+6 k=6-3\sqrt{3}$$

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.

$$\frac{1}{4}k\times 4k+9=12k$$

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

$$kk+9=12k$$

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

$$k^{2}+9=12k$$

Subtract 12k from both sides.

$$k^{2}+9-12k=0$$

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

$$k^{2}-12k=-9$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-12k+\left(-6\right)^{2}=-9+\left(-6\right)^{2}$$

Square -6.

$$k^{2}-12k+36=-9+36$$

Add -9 to 36.

$$k^{2}-12k+36=27$$

Factor k^{2}-12k+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-6\right)^{2}=27$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-6\right)^{2}}=\sqrt{27}$$

Simplify.

$$k-6=3\sqrt{3} k-6=-3\sqrt{3}$$

Add 6 to both sides of the equation.

$$k=3\sqrt{3}+6 k=6-3\sqrt{3}$$