Algebra Examples

$A (4.2 t) = π {(4.2 t)}^{2}$

Step 1

Rewrite using the commutative property of multiplication.

$4.2 A t = π {(4.2 t)}^{2}$

Step 2

Simplify $π {(4.2 t)}^{2}$ .

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the product rule to $4.2 t$ .

$4.2 A t = π ({4.2}^{2} t^{2})$

Step 2.1.2

Raise $4.2$ to the power of $2$ .

$4.2 A t = π (17.64 t^{2})$

Step 2.2

Multiply $17.64$ by $π$ .

$4.2 A t = 55.4176944 t^{2}$

Step 3

Subtract $55.4176944 t^{2}$ from both sides of the equation.

$4.2 A t - 55.4176944 t^{2} = 0$

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 $a = - 55.4176944$ , $b = 4.2 A$ , and $c = 0$ into the quadratic formula and solve for $t$ .

$\frac{- 4.2 A \pm \sqrt{{(4.2 A)}^{2} - 4 \cdot (- 55.4176944 \cdot 0)}}{2 \cdot - 55.4176944}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the product rule to $4.2 A$ .

$t = \frac{- 4.2 A \pm \sqrt{{4.2}^{2} A^{2} - 4 \cdot - 55.4176944 \cdot 0}}{2 \cdot - 55.4176944}$

Step 6.1.2

Raise $4.2$ to the power of $2$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} - 4 \cdot - 55.4176944 \cdot 0}}{2 \cdot - 55.4176944}$

Step 6.1.3

Multiply $- 4 \cdot - 55.4176944 \cdot 0$ .

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Step 6.1.3.1

Multiply $- 4$ by $- 55.4176944$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} + 221.67077763 \cdot 0}}{2 \cdot - 55.4176944}$

Step 6.1.3.2

Multiply $221.67077763$ by $0$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} + 0}}{2 \cdot - 55.4176944}$

Step 6.1.4

Add $17.64 A^{2}$ and $0$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2}}}{2 \cdot - 55.4176944}$

Step 6.1.5

Rewrite $17.64 A^{2}$ as ${(4.2 A)}^{2}$ .

$t = \frac{- 4.2 A \pm \sqrt{{(4.2 A)}^{2}}}{2 \cdot - 55.4176944}$

Step 6.1.6

Pull terms out from under the radical, assuming positive real numbers.

$t = \frac{- 4.2 A \pm 4.2 A}{2 \cdot - 55.4176944}$

Step 6.2

Multiply $2$ by $- 55.4176944$ .

$t = \frac{- 4.2 A \pm 4.2 A}{- 110.83538881}$

Step 6.3

Simplify $\frac{- 4.2 A \pm 4.2 A}{- 110.83538881}$ .

$t = \frac{4.2 A \pm 4.2 A}{110.83538881}$

Step 6.4

Multiply by $1$ .

$t = \frac{1 (4.2 A \pm 4.2 A)}{110.83538881}$

Step 6.5

Factor $110.83538881$ out of $110.83538881$ .

$t = \frac{1 (4.2 A \pm 4.2 A)}{110.83538881 (1)}$

Step 6.6

Separate fractions.

$t = \frac{1}{110.83538881} \cdot \frac{4.2 A \pm 4.2 A}{1}$

Step 6.7

Divide $1$ by $110.83538881$ .

$t = 0.00902238 (\frac{4.2 A \pm 4.2 A}{1})$

Step 6.8

Divide $4.2 A \pm 4.2 A$ by $1$ .

$t = 0.00902238 (4.2 A \pm 4.2 A)$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the product rule to $4.2 A$ .

$t = \frac{- 4.2 A \pm \sqrt{{4.2}^{2} A^{2} - 4 \cdot - 55.4176944 \cdot 0}}{2 \cdot - 55.4176944}$

Step 7.1.2

Raise $4.2$ to the power of $2$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} - 4 \cdot - 55.4176944 \cdot 0}}{2 \cdot - 55.4176944}$

Step 7.1.3

Multiply $- 4 \cdot - 55.4176944 \cdot 0$ .

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Step 7.1.3.1

Multiply $- 4$ by $- 55.4176944$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} + 221.67077763 \cdot 0}}{2 \cdot - 55.4176944}$

Step 7.1.3.2

Multiply $221.67077763$ by $0$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} + 0}}{2 \cdot - 55.4176944}$

Step 7.1.4

Add $17.64 A^{2}$ and $0$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2}}}{2 \cdot - 55.4176944}$

Step 7.1.5

Rewrite $17.64 A^{2}$ as ${(4.2 A)}^{2}$ .

$t = \frac{- 4.2 A \pm \sqrt{{(4.2 A)}^{2}}}{2 \cdot - 55.4176944}$

Step 7.1.6

Pull terms out from under the radical, assuming positive real numbers.

$t = \frac{- 4.2 A \pm 4.2 A}{2 \cdot - 55.4176944}$

Step 7.2

Multiply $2$ by $- 55.4176944$ .

$t = \frac{- 4.2 A \pm 4.2 A}{- 110.83538881}$

Step 7.3

Simplify $\frac{- 4.2 A \pm 4.2 A}{- 110.83538881}$ .

$t = \frac{4.2 A \pm 4.2 A}{110.83538881}$

Step 7.4

Multiply by $1$ .

$t = \frac{1 (4.2 A \pm 4.2 A)}{110.83538881}$

Step 7.5

Factor $110.83538881$ out of $110.83538881$ .

$t = \frac{1 (4.2 A \pm 4.2 A)}{110.83538881 (1)}$

Step 7.6

Separate fractions.

$t = \frac{1}{110.83538881} \cdot \frac{4.2 A \pm 4.2 A}{1}$

Step 7.7

Divide $1$ by $110.83538881$ .

$t = 0.00902238 (\frac{4.2 A \pm 4.2 A}{1})$

Step 7.8

Divide $4.2 A \pm 4.2 A$ by $1$ .

$t = 0.00902238 (4.2 A \pm 4.2 A)$

Step 7.9

Change the $\pm$ to $+$ .

$t = 0.00902238 (4.2 A + 4.2 A)$

Step 7.10

Add $4.2 A$ and $4.2 A$ .

$t = 0.00902238 (8.4 A)$

Step 7.11

Multiply $8.4$ by $0.00902238$ .

$t = 0.07578806 A$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the product rule to $4.2 A$ .

$t = \frac{- 4.2 A \pm \sqrt{{4.2}^{2} A^{2} - 4 \cdot - 55.4176944 \cdot 0}}{2 \cdot - 55.4176944}$

Step 8.1.2

Raise $4.2$ to the power of $2$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} - 4 \cdot - 55.4176944 \cdot 0}}{2 \cdot - 55.4176944}$

Step 8.1.3

Multiply $- 4 \cdot - 55.4176944 \cdot 0$ .

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Step 8.1.3.1

Multiply $- 4$ by $- 55.4176944$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} + 221.67077763 \cdot 0}}{2 \cdot - 55.4176944}$

Step 8.1.3.2

Multiply $221.67077763$ by $0$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2} + 0}}{2 \cdot - 55.4176944}$

Step 8.1.4

Add $17.64 A^{2}$ and $0$ .

$t = \frac{- 4.2 A \pm \sqrt{17.64 A^{2}}}{2 \cdot - 55.4176944}$

Step 8.1.5

Rewrite $17.64 A^{2}$ as ${(4.2 A)}^{2}$ .

$t = \frac{- 4.2 A \pm \sqrt{{(4.2 A)}^{2}}}{2 \cdot - 55.4176944}$

Step 8.1.6

Pull terms out from under the radical, assuming positive real numbers.

$t = \frac{- 4.2 A \pm 4.2 A}{2 \cdot - 55.4176944}$

Step 8.2

Multiply $2$ by $- 55.4176944$ .

$t = \frac{- 4.2 A \pm 4.2 A}{- 110.83538881}$

Step 8.3

Simplify $\frac{- 4.2 A \pm 4.2 A}{- 110.83538881}$ .

$t = \frac{4.2 A \pm 4.2 A}{110.83538881}$

Step 8.4

Multiply by $1$ .

$t = \frac{1 (4.2 A \pm 4.2 A)}{110.83538881}$

Step 8.5

Factor $110.83538881$ out of $110.83538881$ .

$t = \frac{1 (4.2 A \pm 4.2 A)}{110.83538881 (1)}$

Step 8.6

Separate fractions.

$t = \frac{1}{110.83538881} \cdot \frac{4.2 A \pm 4.2 A}{1}$

Step 8.7

Divide $1$ by $110.83538881$ .

$t = 0.00902238 (\frac{4.2 A \pm 4.2 A}{1})$

Step 8.8

Divide $4.2 A \pm 4.2 A$ by $1$ .

$t = 0.00902238 (4.2 A \pm 4.2 A)$

Step 8.9

Change the $\pm$ to $-$ .

$t = 0.00902238 (4.2 A - 4.2 A)$

Step 8.10

Subtract $4.2 A$ from $4.2 A$ .

$t = 0.00902238 (0 A)$

Step 8.11

Multiply $0$ by $A$ .

$t = 0.00902238 \cdot 0$

Step 8.12

Multiply $0.00902238$ by $0$ .

$t = 0$

Step 9

The final answer is the combination of both solutions.

$t = 0.07578806 A$

$t = 0$

Step 10

To rewrite as a function of $A$ , write the equation so that $t$ is by itself on one side of the equal sign and an expression involving only $A$ is on the other side.

$f (A) = 0.07578806 A, 0$