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

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

Solve for k

$$k = \frac{\sqrt{21313} - 97}{24} \approx 2.041238569$$

$$k=\frac{-\sqrt{21313}-97}{24}\approx -10.124571903$$

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), the least common multiple of 4,\left(8-k\right)^{1}.

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

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

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

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

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

To find the opposite of 64-16k+k^{2}, find the opposite of each term.

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

Subtract 64 from 4 to get -60.

$$k-8=-4\left(4k^{2}+8k-60+16k-k^{2}\right)$$

Combine 8k and 16k to get 24k.

$$k-8=-4\left(4k^{2}+24k-60-k^{2}\right)$$

Combine 4k^{2} and -k^{2} to get 3k^{2}.

$$k-8=-4\left(3k^{2}+24k-60\right)$$

Use the distributive property to multiply -4 by 3k^{2}+24k-60.

$$k-8=-12k^{2}-96k+240$$

Add 12k^{2} to both sides.

$$k-8+12k^{2}=-96k+240$$

Add 96k to both sides.

$$k-8+12k^{2}+96k=240$$

Combine k and 96k to get 97k.

$$97k-8+12k^{2}=240$$

Subtract 240 from both sides.

$$97k-8+12k^{2}-240=0$$

Subtract 240 from -8 to get -248.

$$97k-248+12k^{2}=0$$

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.

$$12k^{2}+97k-248=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 97 for b, and -248 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-97±\sqrt{97^{2}-4\times 12\left(-248\right)}}{2\times 12}$$

Square 97.

$$k=\frac{-97±\sqrt{9409-4\times 12\left(-248\right)}}{2\times 12}$$

Multiply -4 times 12.

$$k=\frac{-97±\sqrt{9409-48\left(-248\right)}}{2\times 12}$$

Multiply -48 times -248.

$$k=\frac{-97±\sqrt{9409+11904}}{2\times 12}$$

Add 9409 to 11904.

$$k=\frac{-97±\sqrt{21313}}{2\times 12}$$

Multiply 2 times 12.

$$k=\frac{-97±\sqrt{21313}}{24}$$

Now solve the equation k=\frac{-97±\sqrt{21313}}{24} when ± is plus. Add -97 to \sqrt{21313}.

$$k=\frac{\sqrt{21313}-97}{24}$$

Now solve the equation k=\frac{-97±\sqrt{21313}}{24} when ± is minus. Subtract \sqrt{21313} from -97.

$$k=\frac{-\sqrt{21313}-97}{24}$$

The equation is now solved.

$$k=\frac{\sqrt{21313}-97}{24} k=\frac{-\sqrt{21313}-97}{24}$$

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), the least common multiple of 4,\left(8-k\right)^{1}.

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

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

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

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

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

To find the opposite of 64-16k+k^{2}, find the opposite of each term.

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

Subtract 64 from 4 to get -60.

$$k-8=-4\left(4k^{2}+8k-60+16k-k^{2}\right)$$

Combine 8k and 16k to get 24k.

$$k-8=-4\left(4k^{2}+24k-60-k^{2}\right)$$

Combine 4k^{2} and -k^{2} to get 3k^{2}.

$$k-8=-4\left(3k^{2}+24k-60\right)$$

Use the distributive property to multiply -4 by 3k^{2}+24k-60.

$$k-8=-12k^{2}-96k+240$$

Add 12k^{2} to both sides.

$$k-8+12k^{2}=-96k+240$$

Add 96k to both sides.

$$k-8+12k^{2}+96k=240$$

Combine k and 96k to get 97k.

$$97k-8+12k^{2}=240$$

Add 8 to both sides.

$$97k+12k^{2}=240+8$$

Add 240 and 8 to get 248.

$$97k+12k^{2}=248$$

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.

$$12k^{2}+97k=248$$

Divide both sides by 12.

$$\frac{12k^{2}+97k}{12}=\frac{248}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$k^{2}+\frac{97}{12}k=\frac{248}{12}$$

Reduce the fraction \frac{248}{12} to lowest terms by extracting and canceling out 4.

$$k^{2}+\frac{97}{12}k=\frac{62}{3}$$

Divide \frac{97}{12}, the coefficient of the x term, by 2 to get \frac{97}{24}. Then add the square of \frac{97}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+\frac{97}{12}k+\left(\frac{97}{24}\right)^{2}=\frac{62}{3}+\left(\frac{97}{24}\right)^{2}$$

Square \frac{97}{24} by squaring both the numerator and the denominator of the fraction.

$$k^{2}+\frac{97}{12}k+\frac{9409}{576}=\frac{62}{3}+\frac{9409}{576}$$

Add \frac{62}{3} to \frac{9409}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}+\frac{97}{12}k+\frac{9409}{576}=\frac{21313}{576}$$

Factor k^{2}+\frac{97}{12}k+\frac{9409}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+\frac{97}{24}\right)^{2}=\frac{21313}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+\frac{97}{24}\right)^{2}}=\sqrt{\frac{21313}{576}}$$

Simplify.

$$k+\frac{97}{24}=\frac{\sqrt{21313}}{24} k+\frac{97}{24}=-\frac{\sqrt{21313}}{24}$$

Subtract \frac{97}{24} from both sides of the equation.

$$k=\frac{\sqrt{21313}-97}{24} k=\frac{-\sqrt{21313}-97}{24}$$