Solve 1+30k/8-25/4-18/4-2k^2/4=0

$$1 + \frac { 30 k } { 8 } - \frac { 25 } { 4 } - \frac { 18 } { 4 } - \frac { 2 k ^ { 2 } } { 4 } = 0$$

Solve for k

$$k=\frac{15+\sqrt{87}i}{4}\approx 3.75+2.331844763i$$

$$k=\frac{-\sqrt{87}i+15}{4}\approx 3.75-2.331844763i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 8, the least common multiple of 8,4.

$$8+30k-50-2\times 18-2\times 2k^{2}=0$$

Subtract 50 from 8 to get -42.

$$-42+30k-2\times 18-2\times 2k^{2}=0$$

Multiply -2 and 18 to get -36.

$$-42+30k-36-2\times 2k^{2}=0$$

Subtract 36 from -42 to get -78.

$$-78+30k-2\times 2k^{2}=0$$

Multiply -2 and 2 to get -4.

$$-78+30k-4k^{2}=0$$

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.

$$-4k^{2}+30k-78=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 30 for b, and -78 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-30±\sqrt{30^{2}-4\left(-4\right)\left(-78\right)}}{2\left(-4\right)}$$

Square 30.

$$k=\frac{-30±\sqrt{900-4\left(-4\right)\left(-78\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$k=\frac{-30±\sqrt{900+16\left(-78\right)}}{2\left(-4\right)}$$

Multiply 16 times -78.

$$k=\frac{-30±\sqrt{900-1248}}{2\left(-4\right)}$$

Add 900 to -1248.

$$k=\frac{-30±\sqrt{-348}}{2\left(-4\right)}$$

Take the square root of -348.

$$k=\frac{-30±2\sqrt{87}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$k=\frac{-30±2\sqrt{87}i}{-8}$$

Now solve the equation k=\frac{-30±2\sqrt{87}i}{-8} when ± is plus. Add -30 to 2i\sqrt{87}.

$$k=\frac{-30+2\sqrt{87}i}{-8}$$

Divide -30+2i\sqrt{87} by -8.

$$k=\frac{-\sqrt{87}i+15}{4}$$

Now solve the equation k=\frac{-30±2\sqrt{87}i}{-8} when ± is minus. Subtract 2i\sqrt{87} from -30.

$$k=\frac{-2\sqrt{87}i-30}{-8}$$

Divide -30-2i\sqrt{87} by -8.

$$k=\frac{15+\sqrt{87}i}{4}$$

The equation is now solved.

$$k=\frac{-\sqrt{87}i+15}{4} k=\frac{15+\sqrt{87}i}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 8, the least common multiple of 8,4.

$$8+30k-50-2\times 18-2\times 2k^{2}=0$$

Subtract 50 from 8 to get -42.

$$-42+30k-2\times 18-2\times 2k^{2}=0$$

Multiply -2 and 18 to get -36.

$$-42+30k-36-2\times 2k^{2}=0$$

Subtract 36 from -42 to get -78.

$$-78+30k-2\times 2k^{2}=0$$

Multiply -2 and 2 to get -4.

$$-78+30k-4k^{2}=0$$

Add 78 to both sides. Anything plus zero gives itself.

$$30k-4k^{2}=78$$

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.

$$-4k^{2}+30k=78$$

Divide both sides by -4.

$$\frac{-4k^{2}+30k}{-4}=\frac{78}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$k^{2}+\frac{30}{-4}k=\frac{78}{-4}$$

Reduce the fraction \frac{30}{-4} to lowest terms by extracting and canceling out 2.

$$k^{2}-\frac{15}{2}k=\frac{78}{-4}$$

Reduce the fraction \frac{78}{-4} to lowest terms by extracting and canceling out 2.

$$k^{2}-\frac{15}{2}k=-\frac{39}{2}$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-\frac{15}{2}k+\left(-\frac{15}{4}\right)^{2}=-\frac{39}{2}+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$k^{2}-\frac{15}{2}k+\frac{225}{16}=-\frac{39}{2}+\frac{225}{16}$$

Add -\frac{39}{2} to \frac{225}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}-\frac{15}{2}k+\frac{225}{16}=-\frac{87}{16}$$

Factor k^{2}-\frac{15}{2}k+\frac{225}{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}.

$$\left(k-\frac{15}{4}\right)^{2}=-\frac{87}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-\frac{15}{4}\right)^{2}}=\sqrt{-\frac{87}{16}}$$

Simplify.

$$k-\frac{15}{4}=\frac{\sqrt{87}i}{4} k-\frac{15}{4}=-\frac{\sqrt{87}i}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$k=\frac{15+\sqrt{87}i}{4} k=\frac{-\sqrt{87}i+15}{4}$$