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

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

Solve for k

$$k = \frac{\sqrt{20161} - 95}{24} \approx 1.957893176$$

$$k=\frac{-\sqrt{20161}-95}{24}\approx -9.874559843$$

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}.

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

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

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

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

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

To find the opposite of 4k^{2}+8k+4, find the opposite of each term.

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

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

$$k-8=-4\left(64-16k-3k^{2}-8k-4\right)$$

Combine -16k and -8k to get -24k.

$$k-8=-4\left(64-24k-3k^{2}-4\right)$$

Subtract 4 from 64 to get 60.

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

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

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

Subtract -240 from both sides.

$$k-8-\left(-240\right)=96k+12k^{2}$$

The opposite of -240 is 240.

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

Subtract 96k from both sides.

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

Add -8 and 240 to get 232.

$$k+232-96k=12k^{2}$$

Combine k and -96k to get -95k.

$$-95k+232=12k^{2}$$

Subtract 12k^{2} from both sides.

$$-95k+232-12k^{2}=0$$

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.

$$-12k^{2}-95k+232=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -95 for b, and 232 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-95\right)±\sqrt{\left(-95\right)^{2}-4\left(-12\right)\times 232}}{2\left(-12\right)}$$

Square -95.

$$k=\frac{-\left(-95\right)±\sqrt{9025-4\left(-12\right)\times 232}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$k=\frac{-\left(-95\right)±\sqrt{9025+48\times 232}}{2\left(-12\right)}$$

Multiply 48 times 232.

$$k=\frac{-\left(-95\right)±\sqrt{9025+11136}}{2\left(-12\right)}$$

Add 9025 to 11136.

$$k=\frac{-\left(-95\right)±\sqrt{20161}}{2\left(-12\right)}$$

The opposite of -95 is 95.

$$k=\frac{95±\sqrt{20161}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$k=\frac{95±\sqrt{20161}}{-24}$$

Now solve the equation k=\frac{95±\sqrt{20161}}{-24} when ± is plus. Add 95 to \sqrt{20161}.

$$k=\frac{\sqrt{20161}+95}{-24}$$

Divide 95+\sqrt{20161} by -24.

$$k=\frac{-\sqrt{20161}-95}{24}$$

Now solve the equation k=\frac{95±\sqrt{20161}}{-24} when ± is minus. Subtract \sqrt{20161} from 95.

$$k=\frac{95-\sqrt{20161}}{-24}$$

Divide 95-\sqrt{20161} by -24.

$$k=\frac{\sqrt{20161}-95}{24}$$

The equation is now solved.

$$k=\frac{-\sqrt{20161}-95}{24} k=\frac{\sqrt{20161}-95}{24}$$

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}.

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

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

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

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

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

To find the opposite of 4k^{2}+8k+4, find the opposite of each term.

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

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

$$k-8=-4\left(64-16k-3k^{2}-8k-4\right)$$

Combine -16k and -8k to get -24k.

$$k-8=-4\left(64-24k-3k^{2}-4\right)$$

Subtract 4 from 64 to get 60.

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

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

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

Subtract 96k from both sides.

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

Combine k and -96k to get -95k.

$$-95k-8=-240+12k^{2}$$

Subtract 12k^{2} from both sides.

$$-95k-8-12k^{2}=-240$$

Add 8 to both sides.

$$-95k-12k^{2}=-240+8$$

Add -240 and 8 to get -232.

$$-95k-12k^{2}=-232$$

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.

$$-12k^{2}-95k=-232$$

Divide both sides by -12.

$$\frac{-12k^{2}-95k}{-12}=-\frac{232}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$k^{2}+\left(-\frac{95}{-12}\right)k=-\frac{232}{-12}$$

Divide -95 by -12.

$$k^{2}+\frac{95}{12}k=-\frac{232}{-12}$$

Reduce the fraction \frac{-232}{-12} to lowest terms by extracting and canceling out 4.

$$k^{2}+\frac{95}{12}k=\frac{58}{3}$$

Divide \frac{95}{12}, the coefficient of the x term, by 2 to get \frac{95}{24}. Then add the square of \frac{95}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+\frac{95}{12}k+\left(\frac{95}{24}\right)^{2}=\frac{58}{3}+\left(\frac{95}{24}\right)^{2}$$

Square \frac{95}{24} by squaring both the numerator and the denominator of the fraction.

$$k^{2}+\frac{95}{12}k+\frac{9025}{576}=\frac{58}{3}+\frac{9025}{576}$$

Add \frac{58}{3} to \frac{9025}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}+\frac{95}{12}k+\frac{9025}{576}=\frac{20161}{576}$$

Factor k^{2}+\frac{95}{12}k+\frac{9025}{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}.

$$\left(k+\frac{95}{24}\right)^{2}=\frac{20161}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+\frac{95}{24}\right)^{2}}=\sqrt{\frac{20161}{576}}$$

Simplify.

$$k+\frac{95}{24}=\frac{\sqrt{20161}}{24} k+\frac{95}{24}=-\frac{\sqrt{20161}}{24}$$

Subtract \frac{95}{24} from both sides of the equation.

$$k=\frac{\sqrt{20161}-95}{24} k=\frac{-\sqrt{20161}-95}{24}$$