Solve 16*(-k/2)^2+2[4(-k/2)-3](-k/2)-2k=0

$$16 \times ( \frac { - k } { 2 } ) ^ { 2 } + 2 [ 4 ( \frac { - k } { 2 } ) - 3 ] ( \frac { - k } { 2 } ) - 2 k = 0$$

Solve for k

$$k=-\frac{1}{6}\approx -0.166666667$$

$$k=0$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$32\times \left(\frac{-k}{2}\right)^{2}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

To raise \frac{-k}{2} to a power, raise both numerator and denominator to the power and then divide.

$$32\times \frac{\left(-k\right)^{2}}{2^{2}}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

Express 32\times \frac{\left(-k\right)^{2}}{2^{2}} as a single fraction.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+2\left(2\left(-k\right)-3\right)\left(-k\right)-4k=0$$

Use the distributive property to multiply 2 by 2\left(-k\right)-3.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\left(4\left(-k\right)-6\right)\left(-k\right)-4k=0$$

Use the distributive property to multiply 4\left(-k\right)-6 by -k.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4\left(-k\right)^{2}-6\left(-k\right)-4k=0$$

Calculate -k to the power of 2 and get k^{2}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}-6\left(-k\right)-4k=0$$

Multiply -6 and -1 to get 6.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}+6k-4k=0$$

Combine 6k and -4k to get 2k.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}+2k=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4k^{2}+2k times \frac{2^{2}}{2^{2}}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Since \frac{32\left(-k\right)^{2}}{2^{2}} and \frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{32\left(-k\right)^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4k^{2}+2k times \frac{2^{2}}{2^{2}}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Since \frac{32\left(-k\right)^{2}}{2^{2}} and \frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{32\left(-k\right)^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Calculate -k to the power of 2 and get k^{2}.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Calculate 2 to the power of 2 and get 4.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 4}{2^{2}}=0$$

Calculate 2 to the power of 2 and get 4.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 4}{4}=0$$

Divide each term of 32k^{2}+\left(4k^{2}+2k\right)\times 4 by 4 to get 8k^{2}+4k^{2}+2k.

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

Combine 8k^{2} and 4k^{2} to get 12k^{2}.

$$12k^{2}+2k=0$$

Factor out k.

$$k\left(12k+2\right)=0$$

To find equation solutions, solve k=0 and 12k+2=0.

$$k=0 k=-\frac{1}{6}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$32\times \left(\frac{-k}{2}\right)^{2}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

To raise \frac{-k}{2} to a power, raise both numerator and denominator to the power and then divide.

$$32\times \frac{\left(-k\right)^{2}}{2^{2}}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

Express 32\times \frac{\left(-k\right)^{2}}{2^{2}} as a single fraction.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+2\left(2\left(-k\right)-3\right)\left(-k\right)-4k=0$$

Use the distributive property to multiply 2 by 2\left(-k\right)-3.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\left(4\left(-k\right)-6\right)\left(-k\right)-4k=0$$

Use the distributive property to multiply 4\left(-k\right)-6 by -k.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4\left(-k\right)^{2}-6\left(-k\right)-4k=0$$

Calculate -k to the power of 2 and get k^{2}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}-6\left(-k\right)-4k=0$$

Multiply -6 and -1 to get 6.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}+6k-4k=0$$

Combine 6k and -4k to get 2k.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}+2k=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4k^{2}+2k times \frac{2^{2}}{2^{2}}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Since \frac{32\left(-k\right)^{2}}{2^{2}} and \frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{32\left(-k\right)^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Calculate -k to the power of 2 and get k^{2}.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Calculate 2 to the power of 2 and get 4.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 4}{2^{2}}=0$$

Calculate 2 to the power of 2 and get 4.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 4}{4}=0$$

Divide each term of 32k^{2}+\left(4k^{2}+2k\right)\times 4 by 4 to get 8k^{2}+4k^{2}+2k.

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

Combine 8k^{2} and 4k^{2} to get 12k^{2}.

$$12k^{2}+2k=0$$

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

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

Take the square root of 2^{2}.

$$k=\frac{-2±2}{2\times 12}$$

Multiply 2 times 12.

$$k=\frac{-2±2}{24}$$

Now solve the equation k=\frac{-2±2}{24} when ± is plus. Add -2 to 2.

$$k=\frac{0}{24}$$

Divide 0 by 24.

$$k=0$$

Now solve the equation k=\frac{-2±2}{24} when ± is minus. Subtract 2 from -2.

$$k=-\frac{4}{24}$$

Reduce the fraction \frac{-4}{24} to lowest terms by extracting and canceling out 4.

$$k=-\frac{1}{6}$$

The equation is now solved.

$$k=0 k=-\frac{1}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$32\times \left(\frac{-k}{2}\right)^{2}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

To raise \frac{-k}{2} to a power, raise both numerator and denominator to the power and then divide.

$$32\times \frac{\left(-k\right)^{2}}{2^{2}}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

Express 32\times \frac{\left(-k\right)^{2}}{2^{2}} as a single fraction.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+2\left(4\times \frac{-k}{2}-3\right)\left(-k\right)-4k=0$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+2\left(2\left(-k\right)-3\right)\left(-k\right)-4k=0$$

Use the distributive property to multiply 2 by 2\left(-k\right)-3.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\left(4\left(-k\right)-6\right)\left(-k\right)-4k=0$$

Use the distributive property to multiply 4\left(-k\right)-6 by -k.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4\left(-k\right)^{2}-6\left(-k\right)-4k=0$$

Calculate -k to the power of 2 and get k^{2}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}-6\left(-k\right)-4k=0$$

Multiply -6 and -1 to get 6.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}+6k-4k=0$$

Combine 6k and -4k to get 2k.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+4k^{2}+2k=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4k^{2}+2k times \frac{2^{2}}{2^{2}}.

$$\frac{32\left(-k\right)^{2}}{2^{2}}+\frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Since \frac{32\left(-k\right)^{2}}{2^{2}} and \frac{\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{32\left(-k\right)^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Calculate -k to the power of 2 and get k^{2}.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 2^{2}}{2^{2}}=0$$

Calculate 2 to the power of 2 and get 4.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 4}{2^{2}}=0$$

Calculate 2 to the power of 2 and get 4.

$$\frac{32k^{2}+\left(4k^{2}+2k\right)\times 4}{4}=0$$

Divide each term of 32k^{2}+\left(4k^{2}+2k\right)\times 4 by 4 to get 8k^{2}+4k^{2}+2k.

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

Combine 8k^{2} and 4k^{2} to get 12k^{2}.

$$12k^{2}+2k=0$$

Divide both sides by 12.

$$\frac{12k^{2}+2k}{12}=\frac{0}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$k^{2}+\frac{2}{12}k=\frac{0}{12}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$k^{2}+\frac{1}{6}k=\frac{0}{12}$$

Divide 0 by 12.

$$k^{2}+\frac{1}{6}k=0$$

Divide \frac{1}{6}, the coefficient of the x term, by 2 to get \frac{1}{12}. Then add the square of \frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+\frac{1}{6}k+\left(\frac{1}{12}\right)^{2}=\left(\frac{1}{12}\right)^{2}$$

Square \frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$k^{2}+\frac{1}{6}k+\frac{1}{144}=\frac{1}{144}$$

Factor k^{2}+\frac{1}{6}k+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+\frac{1}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+\frac{1}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$k+\frac{1}{12}=\frac{1}{12} k+\frac{1}{12}=-\frac{1}{12}$$

Subtract \frac{1}{12} from both sides of the equation.

$$k=0 k=-\frac{1}{6}$$