Solve 3-2k/2-k+4-2k/1+k=4

$$\frac { 3 - 2 k } { 2 - k } + \frac { 4 - 2 k } { 1 + k } = 4$$

Solve for k

$$k = \frac{\sqrt{73} + 11}{8} \approx 2.443000468$$

$$k=\frac{11-\sqrt{73}}{8}\approx 0.306999532$$

Steps Using the Quadratic Formula

Variable k cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(k-2\right)\left(k+1\right), the least common multiple of 2-k,1+k.

$$-\left(1+k\right)\left(3-2k\right)+\left(k-2\right)\left(4-2k\right)=4\left(k-2\right)\left(k+1\right)$$

To find the opposite of 1+k, find the opposite of each term.

$$\left(-1-k\right)\left(3-2k\right)+\left(k-2\right)\left(4-2k\right)=4\left(k-2\right)\left(k+1\right)$$

Use the distributive property to multiply -1-k by 3-2k and combine like terms.

$$-3-k+2k^{2}+\left(k-2\right)\left(4-2k\right)=4\left(k-2\right)\left(k+1\right)$$

Use the distributive property to multiply k-2 by 4-2k and combine like terms.

$$-3-k+2k^{2}+8k-2k^{2}-8=4\left(k-2\right)\left(k+1\right)$$

Combine -k and 8k to get 7k.

$$-3+7k+2k^{2}-2k^{2}-8=4\left(k-2\right)\left(k+1\right)$$

Combine 2k^{2} and -2k^{2} to get 0.

$$-3+7k-8=4\left(k-2\right)\left(k+1\right)$$

Subtract 8 from -3 to get -11.

$$-11+7k=4\left(k-2\right)\left(k+1\right)$$

Use the distributive property to multiply 4 by k-2.

$$-11+7k=\left(4k-8\right)\left(k+1\right)$$

Use the distributive property to multiply 4k-8 by k+1 and combine like terms.

$$-11+7k=4k^{2}-4k-8$$

Subtract 4k^{2} from both sides.

$$-11+7k-4k^{2}=-4k-8$$

Add 4k to both sides.

$$-11+7k-4k^{2}+4k=-8$$

Combine 7k and 4k to get 11k.

$$-11+11k-4k^{2}=-8$$

Add 8 to both sides.

$$-11+11k-4k^{2}+8=0$$

Add -11 and 8 to get -3.

$$-3+11k-4k^{2}=0$$

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.

$$-4k^{2}+11k-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 11 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-11±\sqrt{11^{2}-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Square 11.

$$k=\frac{-11±\sqrt{121-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$k=\frac{-11±\sqrt{121+16\left(-3\right)}}{2\left(-4\right)}$$

Multiply 16 times -3.

$$k=\frac{-11±\sqrt{121-48}}{2\left(-4\right)}$$

Add 121 to -48.

$$k=\frac{-11±\sqrt{73}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$k=\frac{-11±\sqrt{73}}{-8}$$

Now solve the equation k=\frac{-11±\sqrt{73}}{-8} when ± is plus. Add -11 to \sqrt{73}.

$$k=\frac{\sqrt{73}-11}{-8}$$

Divide -11+\sqrt{73} by -8.

$$k=\frac{11-\sqrt{73}}{8}$$

Now solve the equation k=\frac{-11±\sqrt{73}}{-8} when ± is minus. Subtract \sqrt{73} from -11.

$$k=\frac{-\sqrt{73}-11}{-8}$$

Divide -11-\sqrt{73} by -8.

$$k=\frac{\sqrt{73}+11}{8}$$

The equation is now solved.

$$k=\frac{11-\sqrt{73}}{8} k=\frac{\sqrt{73}+11}{8}$$

Steps for Completing the Square

Variable k cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(k-2\right)\left(k+1\right), the least common multiple of 2-k,1+k.

$$-\left(1+k\right)\left(3-2k\right)+\left(k-2\right)\left(4-2k\right)=4\left(k-2\right)\left(k+1\right)$$

To find the opposite of 1+k, find the opposite of each term.

$$\left(-1-k\right)\left(3-2k\right)+\left(k-2\right)\left(4-2k\right)=4\left(k-2\right)\left(k+1\right)$$

Use the distributive property to multiply -1-k by 3-2k and combine like terms.

$$-3-k+2k^{2}+\left(k-2\right)\left(4-2k\right)=4\left(k-2\right)\left(k+1\right)$$

Use the distributive property to multiply k-2 by 4-2k and combine like terms.

$$-3-k+2k^{2}+8k-2k^{2}-8=4\left(k-2\right)\left(k+1\right)$$

Combine -k and 8k to get 7k.

$$-3+7k+2k^{2}-2k^{2}-8=4\left(k-2\right)\left(k+1\right)$$

Combine 2k^{2} and -2k^{2} to get 0.

$$-3+7k-8=4\left(k-2\right)\left(k+1\right)$$

Subtract 8 from -3 to get -11.

$$-11+7k=4\left(k-2\right)\left(k+1\right)$$

Use the distributive property to multiply 4 by k-2.

$$-11+7k=\left(4k-8\right)\left(k+1\right)$$

Use the distributive property to multiply 4k-8 by k+1 and combine like terms.

$$-11+7k=4k^{2}-4k-8$$

Subtract 4k^{2} from both sides.

$$-11+7k-4k^{2}=-4k-8$$

Add 4k to both sides.

$$-11+7k-4k^{2}+4k=-8$$

Combine 7k and 4k to get 11k.

$$-11+11k-4k^{2}=-8$$

Add 11 to both sides.

$$11k-4k^{2}=-8+11$$

Add -8 and 11 to get 3.

$$11k-4k^{2}=3$$

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.

$$-4k^{2}+11k=3$$

Divide both sides by -4.

$$\frac{-4k^{2}+11k}{-4}=\frac{3}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$k^{2}+\frac{11}{-4}k=\frac{3}{-4}$$

Divide 11 by -4.

$$k^{2}-\frac{11}{4}k=\frac{3}{-4}$$

Divide 3 by -4.

$$k^{2}-\frac{11}{4}k=-\frac{3}{4}$$

Divide -\frac{11}{4}, the coefficient of the x term, by 2 to get -\frac{11}{8}. Then add the square of -\frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-\frac{11}{4}k+\left(-\frac{11}{8}\right)^{2}=-\frac{3}{4}+\left(-\frac{11}{8}\right)^{2}$$

Square -\frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$k^{2}-\frac{11}{4}k+\frac{121}{64}=-\frac{3}{4}+\frac{121}{64}$$

Add -\frac{3}{4} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}-\frac{11}{4}k+\frac{121}{64}=\frac{73}{64}$$

Factor k^{2}-\frac{11}{4}k+\frac{121}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-\frac{11}{8}\right)^{2}=\frac{73}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-\frac{11}{8}\right)^{2}}=\sqrt{\frac{73}{64}}$$

Simplify.

$$k-\frac{11}{8}=\frac{\sqrt{73}}{8} k-\frac{11}{8}=-\frac{\sqrt{73}}{8}$$

Add \frac{11}{8} to both sides of the equation.

$$k=\frac{\sqrt{73}+11}{8} k=\frac{11-\sqrt{73}}{8}$$