Solve 5k+3/k+12=k/4+19/4k+48

$$\frac { 5 k + 3 } { k + 12 } = \frac { k } { 4 } + \frac { 19 } { 4 k + 48 }$$

Solve for k

$$k=1$$

$$k=7$$

Steps Using the Quadratic Formula

Variable k cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by 4\left(k+12\right), the least common multiple of k+12,4,4k+48.

$$4\left(5k+3\right)=\left(k+12\right)k+19$$

Use the distributive property to multiply 4 by 5k+3.

$$20k+12=\left(k+12\right)k+19$$

Use the distributive property to multiply k+12 by k.

$$20k+12=k^{2}+12k+19$$

Subtract k^{2} from both sides.

$$20k+12-k^{2}=12k+19$$

Subtract 12k from both sides.

$$20k+12-k^{2}-12k=19$$

Combine 20k and -12k to get 8k.

$$8k+12-k^{2}=19$$

Subtract 19 from both sides.

$$8k+12-k^{2}-19=0$$

Subtract 19 from 12 to get -7.

$$8k-7-k^{2}=0$$

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.

$$-k^{2}+8k-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Square 8.

$$k=\frac{-8±\sqrt{64-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$k=\frac{-8±\sqrt{64+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$k=\frac{-8±\sqrt{64-28}}{2\left(-1\right)}$$

Add 64 to -28.

$$k=\frac{-8±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$k=\frac{-8±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$k=\frac{-8±6}{-2}$$

Now solve the equation k=\frac{-8±6}{-2} when ± is plus. Add -8 to 6.

$$k=-\frac{2}{-2}$$

Divide -2 by -2.

$$k=1$$

Now solve the equation k=\frac{-8±6}{-2} when ± is minus. Subtract 6 from -8.

$$k=-\frac{14}{-2}$$

Divide -14 by -2.

$$k=7$$

The equation is now solved.

$$k=1 k=7$$

Steps for Completing the Square

Variable k cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by 4\left(k+12\right), the least common multiple of k+12,4,4k+48.

$$4\left(5k+3\right)=\left(k+12\right)k+19$$

Use the distributive property to multiply 4 by 5k+3.

$$20k+12=\left(k+12\right)k+19$$

Use the distributive property to multiply k+12 by k.

$$20k+12=k^{2}+12k+19$$

Subtract k^{2} from both sides.

$$20k+12-k^{2}=12k+19$$

Subtract 12k from both sides.

$$20k+12-k^{2}-12k=19$$

Combine 20k and -12k to get 8k.

$$8k+12-k^{2}=19$$

Subtract 12 from both sides.

$$8k-k^{2}=19-12$$

Subtract 12 from 19 to get 7.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -1.

$$\frac{-k^{2}+8k}{-1}=\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$k^{2}+\frac{8}{-1}k=\frac{7}{-1}$$

Divide 8 by -1.

$$k^{2}-8k=\frac{7}{-1}$$

Divide 7 by -1.

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

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.

$$k^{2}-8k+\left(-4\right)^{2}=-7+\left(-4\right)^{2}$$

Square -4.

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

Add -7 to 16.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$k-4=3 k-4=-3$$

Add 4 to both sides of the equation.

$$k=7 k=1$$