Solve 1/4=(2k+2)^2-(8-k)/(8-k)^2

$$\frac { 1 } { 4 } = \frac { ( 2 k + 2 ) ^ { 2 } - ( 8 - k ) } { ( 8 - k ) ^ { 2 } }$$

Solve for k

$$k = \frac{2 \sqrt{469} - 26}{15} \approx 1.15418771$$

$$k=\frac{-2\sqrt{469}-26}{15}\approx -4.620854377$$

Steps Using the Quadratic Formula

Variable k cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by 4\left(k-8\right)^{2}, the least common multiple of 4,\left(8-k\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(k-8\right)^{2}.

$$k^{2}-16k+64=4\left(\left(2k+2\right)^{2}-\left(8-k\right)\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2k+2\right)^{2}.

$$k^{2}-16k+64=4\left(4k^{2}+8k+4-\left(8-k\right)\right)$$

To find the opposite of 8-k, find the opposite of each term.

$$k^{2}-16k+64=4\left(4k^{2}+8k+4-8+k\right)$$

Subtract 8 from 4 to get -4.

$$k^{2}-16k+64=4\left(4k^{2}+8k-4+k\right)$$

Combine 8k and k to get 9k.

$$k^{2}-16k+64=4\left(4k^{2}+9k-4\right)$$

Use the distributive property to multiply 4 by 4k^{2}+9k-4.

$$k^{2}-16k+64=16k^{2}+36k-16$$

Subtract 16k^{2} from both sides.

$$k^{2}-16k+64-16k^{2}=36k-16$$

Combine k^{2} and -16k^{2} to get -15k^{2}.

$$-15k^{2}-16k+64=36k-16$$

Subtract 36k from both sides.

$$-15k^{2}-16k+64-36k=-16$$

Combine -16k and -36k to get -52k.

$$-15k^{2}-52k+64=-16$$

Add 16 to both sides.

$$-15k^{2}-52k+64+16=0$$

Add 64 and 16 to get 80.

$$-15k^{2}-52k+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, -52 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\left(-15\right)\times 80}}{2\left(-15\right)}$$

Square -52.

$$k=\frac{-\left(-52\right)±\sqrt{2704-4\left(-15\right)\times 80}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$k=\frac{-\left(-52\right)±\sqrt{2704+60\times 80}}{2\left(-15\right)}$$

Multiply 60 times 80.

$$k=\frac{-\left(-52\right)±\sqrt{2704+4800}}{2\left(-15\right)}$$

Add 2704 to 4800.

$$k=\frac{-\left(-52\right)±\sqrt{7504}}{2\left(-15\right)}$$

Take the square root of 7504.

$$k=\frac{-\left(-52\right)±4\sqrt{469}}{2\left(-15\right)}$$

The opposite of -52 is 52.

$$k=\frac{52±4\sqrt{469}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$k=\frac{52±4\sqrt{469}}{-30}$$

Now solve the equation k=\frac{52±4\sqrt{469}}{-30} when ± is plus. Add 52 to 4\sqrt{469}.

$$k=\frac{4\sqrt{469}+52}{-30}$$

Divide 52+4\sqrt{469} by -30.

$$k=\frac{-2\sqrt{469}-26}{15}$$

Now solve the equation k=\frac{52±4\sqrt{469}}{-30} when ± is minus. Subtract 4\sqrt{469} from 52.

$$k=\frac{52-4\sqrt{469}}{-30}$$

Divide 52-4\sqrt{469} by -30.

$$k=\frac{2\sqrt{469}-26}{15}$$

The equation is now solved.

$$k=\frac{-2\sqrt{469}-26}{15} k=\frac{2\sqrt{469}-26}{15}$$

Steps for Completing the Square

Variable k cannot be equal to 8 since division by zero is not defined. Multiply both sides of the equation by 4\left(k-8\right)^{2}, the least common multiple of 4,\left(8-k\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(k-8\right)^{2}.

$$k^{2}-16k+64=4\left(\left(2k+2\right)^{2}-\left(8-k\right)\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2k+2\right)^{2}.

$$k^{2}-16k+64=4\left(4k^{2}+8k+4-\left(8-k\right)\right)$$

To find the opposite of 8-k, find the opposite of each term.

$$k^{2}-16k+64=4\left(4k^{2}+8k+4-8+k\right)$$

Subtract 8 from 4 to get -4.

$$k^{2}-16k+64=4\left(4k^{2}+8k-4+k\right)$$

Combine 8k and k to get 9k.

$$k^{2}-16k+64=4\left(4k^{2}+9k-4\right)$$

Use the distributive property to multiply 4 by 4k^{2}+9k-4.

$$k^{2}-16k+64=16k^{2}+36k-16$$

Subtract 16k^{2} from both sides.

$$k^{2}-16k+64-16k^{2}=36k-16$$

Combine k^{2} and -16k^{2} to get -15k^{2}.

$$-15k^{2}-16k+64=36k-16$$

Subtract 36k from both sides.

$$-15k^{2}-16k+64-36k=-16$$

Combine -16k and -36k to get -52k.

$$-15k^{2}-52k+64=-16$$

Subtract 64 from both sides.

$$-15k^{2}-52k=-16-64$$

Subtract 64 from -16 to get -80.

$$-15k^{2}-52k=-80$$

Divide both sides by -15.

$$\frac{-15k^{2}-52k}{-15}=-\frac{80}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$k^{2}+\left(-\frac{52}{-15}\right)k=-\frac{80}{-15}$$

Divide -52 by -15.

$$k^{2}+\frac{52}{15}k=-\frac{80}{-15}$$

Reduce the fraction \frac{-80}{-15} to lowest terms by extracting and canceling out 5.

$$k^{2}+\frac{52}{15}k=\frac{16}{3}$$

Divide \frac{52}{15}, the coefficient of the x term, by 2 to get \frac{26}{15}. Then add the square of \frac{26}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+\frac{52}{15}k+\left(\frac{26}{15}\right)^{2}=\frac{16}{3}+\left(\frac{26}{15}\right)^{2}$$

Square \frac{26}{15} by squaring both the numerator and the denominator of the fraction.

$$k^{2}+\frac{52}{15}k+\frac{676}{225}=\frac{16}{3}+\frac{676}{225}$$

Add \frac{16}{3} to \frac{676}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}+\frac{52}{15}k+\frac{676}{225}=\frac{1876}{225}$$

Factor k^{2}+\frac{52}{15}k+\frac{676}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+\frac{26}{15}\right)^{2}=\frac{1876}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+\frac{26}{15}\right)^{2}}=\sqrt{\frac{1876}{225}}$$

Simplify.

$$k+\frac{26}{15}=\frac{2\sqrt{469}}{15} k+\frac{26}{15}=-\frac{2\sqrt{469}}{15}$$

Subtract \frac{26}{15} from both sides of the equation.

$$k=\frac{2\sqrt{469}-26}{15} k=\frac{-2\sqrt{469}-26}{15}$$