Algebra Examples

$k \cdot 0.8 k + 2 m = 3 m + k$

Step 1

Simplify each term.

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Step 1.1

Rewrite using the commutative property of multiplication.

$0.8 k \cdot k + 2 m = 3 m + k$

Step 1.2

Multiply $k$ by $k$ by adding the exponents.

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Step 1.2.1

Move $k$ .

$0.8 (k \cdot k) + 2 m = 3 m + k$

Step 1.2.2

Multiply $k$ by $k$ .

$0.8 k^{2} + 2 m = 3 m + k$

Step 2

Subtract $k$ from both sides of the equation.

$0.8 k^{2} + 2 m - k = 3 m$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Subtract $3 m$ from both sides of the equation.

$0.8 k^{2} + 2 m - k - 3 m = 0$

Step 3.2

Subtract $3 m$ from $2 m$ .

$0.8 k^{2} - k - m = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 0.8$ , $b = - 1$ , and $c = - m$ into the quadratic formula and solve for $k$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (0.8 \cdot (- m))}}{2 \cdot 0.8}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $- 1$ out of ${(- 1)}^{2} - 4 \cdot 0.8 \cdot - 1 m$ .

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Step 6.1.1.1

Factor $- 1$ out of ${(- 1)}^{2}$ .

$k = \frac{1 \pm \sqrt{1 - 4 \cdot 0.8 \cdot (- 1 m)}}{2 \cdot 0.8}$

Step 6.1.1.2

Factor $- 1$ out of $- 4 \cdot 0.8 \cdot - 1 m$ .

$k = \frac{1 \pm \sqrt{1 - (- 4 \cdot (0.8 m))}}{2 \cdot 0.8}$

Step 6.1.1.3

Factor $- 1$ out of $- - 1 - (- 4 \cdot 0.8 m)$ .

$k = \frac{1 \pm \sqrt{- (- 1 - 4 \cdot (0.8 m))}}{2 \cdot 0.8}$

Step 6.1.2

Multiply $- 4$ by $0.8$ .

$k = \frac{1 \pm \sqrt{- (- 1 - 3.2 m)}}{2 \cdot 0.8}$

Step 6.1.3

Factor $0.2$ out of $- 1 - 3.2 m$ .

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Step 6.1.3.1

Factor $0.2$ out of $- 1$ .

$k = \frac{1 \pm \sqrt{- (0.2 (- 5) - 3.2 m)}}{2 \cdot 0.8}$

Step 6.1.3.2

Factor $0.2$ out of $- 3.2 m$ .

$k = \frac{1 \pm \sqrt{- (0.2 (- 5) + 0.2 (- 16 m))}}{2 \cdot 0.8}$

Step 6.1.3.3

Factor $0.2$ out of $0.2 (- 5) + 0.2 (- 16 m)$ .

$k = \frac{1 \pm \sqrt{- (0.2 (- 5 - 16 m))}}{2 \cdot 0.8}$

$k = \frac{1 \pm \sqrt{- 1 \cdot (0.2 (- 5 - 16 m))}}{2 \cdot 0.8}$

Step 6.1.4

Multiply $- 1$ by $0.2$ .

$k = \frac{1 \pm \sqrt{- 0.2 (- 5 - 16 m)}}{2 \cdot 0.8}$

Step 6.1.5

Rewrite $- 0.2 (- 5 - 16 m)$ as ${({0.6687403}^{2})}^{2} \cdot (- 1 (- 5 - 16 m))$ .

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Step 6.1.5.1

Factor $0.2$ out of $- 0.2$ .

$k = \frac{1 \pm \sqrt{0.2 (- 1) (- 5 - 16 m)}}{2 \cdot 0.8}$

Step 6.1.5.2

Rewrite $0.2$ as ${0.44721359}^{2}$ .

$k = \frac{1 \pm \sqrt{{0.44721359}^{2} \cdot (- 1 (- 5 - 16 m))}}{2 \cdot 0.8}$

Step 6.1.5.3

Rewrite $0.44721359$ as ${0.6687403}^{2}$ .

$k = \frac{1 \pm \sqrt{{({0.6687403}^{2})}^{2} \cdot (- 1 (- 5 - 16 m))}}{2 \cdot 0.8}$

Step 6.1.5.4

Add parentheses.

$k = \frac{1 \pm \sqrt{{({0.6687403}^{2})}^{2} \cdot (- 1 (- 5 - 16 m))}}{2 \cdot 0.8}$

Step 6.1.6

Pull terms out from under the radical.

$k = \frac{1 \pm {0.6687403}^{2} \sqrt{- 1 (- 5 - 16 m)}}{2 \cdot 0.8}$

Step 6.1.7

Raise $0.6687403$ to the power of $2$ .

$k = \frac{1 \pm 0.44721359 \sqrt{- 1 (- 5 - 16 m)}}{2 \cdot 0.8}$

Step 6.2

Multiply $2$ by $0.8$ .

$k = \frac{1 \pm 0.44721359 \sqrt{- 1 (- 5 - 16 m)}}{1.6}$

Step 6.3

Simplify $\frac{1 \pm 0.44721359 \sqrt{- 1 (- 5 - 16 m)}}{1.6}$ .

$k = \frac{5 \pm 2.23606797 \sqrt{- 1 (- 5 - 16 m)}}{8}$

Step 7

The final answer is the combination of both solutions.

$k = \frac{5 + 2.23606797 \sqrt{- 1 (- 5 - 16 m)}}{8}$

$k = \frac{5 - 2.23606797 \sqrt{- 1 (- 5 - 16 m)}}{8}$