Algebra Examples

$a \cdot 0.7 a - 2 b = 5 a + b$

Step 1

Simplify each term.

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Step 1.1

Rewrite using the commutative property of multiplication.

$0.7 a \cdot a - 2 b = 5 a + b$

Step 1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 1.2.1

Move $a$ .

$0.7 (a \cdot a) - 2 b = 5 a + b$

Step 1.2.2

Multiply $a$ by $a$ .

$0.7 a^{2} - 2 b = 5 a + b$

Step 2

Subtract $5 a$ from both sides of the equation.

$0.7 a^{2} - 2 b - 5 a = b$

Step 3

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

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Step 3.1

Subtract $b$ from both sides of the equation.

$0.7 a^{2} - 2 b - 5 a - b = 0$

Step 3.2

Subtract $b$ from $- 2 b$ .

$0.7 a^{2} - 5 a - 3 b = 0$

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 $a = 0.7$ , $b = - 5$ , and $c = - 3 b$ into the quadratic formula and solve for $a$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (0.7 \cdot (- 3 b))}}{2 \cdot 0.7}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 5$ to the power of $2$ .

$a = \frac{5 \pm \sqrt{25 - 4 \cdot 0.7 \cdot (- 3 b)}}{2 \cdot 0.7}$

Step 6.1.2

Multiply $- 4 \cdot 0.7 \cdot - 3$ .

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Step 6.1.2.1

Multiply $- 4$ by $0.7$ .

$a = \frac{5 \pm \sqrt{25 - 2.8 \cdot (- 3 b)}}{2 \cdot 0.7}$

Step 6.1.2.2

Multiply $- 2.8$ by $- 3$ .

$a = \frac{5 \pm \sqrt{25 + 8.4 b}}{2 \cdot 0.7}$

Step 6.1.3

Factor $0.2$ out of $25 + 8.4 b$ .

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Step 6.1.3.1

Factor $0.2$ out of $25$ .

$a = \frac{5 \pm \sqrt{0.2 (125) + 8.4 b}}{2 \cdot 0.7}$

Step 6.1.3.2

Factor $0.2$ out of $8.4 b$ .

$a = \frac{5 \pm \sqrt{0.2 (125) + 0.2 (42 b)}}{2 \cdot 0.7}$

Step 6.1.3.3

Factor $0.2$ out of $0.2 (125) + 0.2 (42 b)$ .

$a = \frac{5 \pm \sqrt{0.2 (125 + 42 b)}}{2 \cdot 0.7}$

Step 6.1.4

Rewrite $0.2 (125 + 42 b)$ as ${({0.6687403}^{2})}^{2} (125 + 42 b)$ .

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Step 6.1.4.1

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

$a = \frac{5 \pm \sqrt{{0.44721359}^{2} (125 + 42 b)}}{2 \cdot 0.7}$

Step 6.1.4.2

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

$a = \frac{5 \pm \sqrt{{({0.6687403}^{2})}^{2} (125 + 42 b)}}{2 \cdot 0.7}$

Step 6.1.5

Pull terms out from under the radical.

$a = \frac{5 \pm {0.6687403}^{2} \sqrt{125 + 42 b}}{2 \cdot 0.7}$

Step 6.1.6

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

$a = \frac{5 \pm 0.44721359 \sqrt{125 + 42 b}}{2 \cdot 0.7}$

Step 6.2

Multiply $2$ by $0.7$ .

$a = \frac{5 \pm 0.44721359 \sqrt{125 + 42 b}}{1.4}$

Step 6.3

Simplify $\frac{5 \pm 0.44721359 \sqrt{125 + 42 b}}{1.4}$ .

$a = \frac{25 \pm 2.23606797 \sqrt{125 + 42 b}}{7}$

Step 7

The final answer is the combination of both solutions.

$a = \frac{25 + 2.23606797 \sqrt{125 + 42 b}}{7}$

$a = \frac{25 - 2.23606797 \sqrt{125 + 42 b}}{7}$