Basic Math Examples

$4 a^{4} = 8 a^{3} + 4 a^{2}$

Step 1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$8 a^{3} + 4 a^{2} = 4 a^{4}$

Step 2

Subtract $4 a^{4}$ from both sides of the equation.

$8 a^{3} + 4 a^{2} - 4 a^{4} = 0$

Step 3

Factor $- 4 a^{2}$ out of $8 a^{3} + 4 a^{2} - 4 a^{4}$ .

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Step 3.1

Reorder the expression.

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Step 3.1.1

Move $4 a^{2}$ .

$8 a^{3} - 4 a^{4} + 4 a^{2} = 0$

Step 3.1.2

Reorder $8 a^{3}$ and $- 4 a^{4}$ .

$- 4 a^{4} + 8 a^{3} + 4 a^{2} = 0$

Step 3.2

Factor $- 4 a^{2}$ out of $- 4 a^{4}$ .

$- 4 a^{2} a^{2} + 8 a^{3} + 4 a^{2} = 0$

Step 3.3

Factor $- 4 a^{2}$ out of $8 a^{3}$ .

$- 4 a^{2} a^{2} - 4 a^{2} (- 2 a) + 4 a^{2} = 0$

Step 3.4

Factor $- 4 a^{2}$ out of $4 a^{2}$ .

$- 4 a^{2} a^{2} - 4 a^{2} (- 2 a) - 4 a^{2} \cdot - 1 = 0$

Step 3.5

Factor $- 4 a^{2}$ out of $- 4 a^{2} (a^{2}) - 4 a^{2} (- 2 a)$ .

$- 4 a^{2} (a^{2} - 2 a) - 4 a^{2} \cdot - 1 = 0$

Step 3.6

Factor $- 4 a^{2}$ out of $- 4 a^{2} (a^{2} - 2 a) - 4 a^{2} (- 1)$ .

$- 4 a^{2} (a^{2} - 2 a - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a^{2} = 0$

$a^{2} - 2 a - 1 = 0$

Step 5

Set $a^{2}$ equal to $0$ and solve for $a$ .

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Step 5.1

Set $a^{2}$ equal to $0$ .

$a^{2} = 0$

Step 5.2

Solve $a^{2} = 0$ for $a$ .

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Step 5.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{0}$

Step 5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.2.2.1

Rewrite $0$ as $0^{2}$ .

$a = \pm \sqrt{0^{2}}$

Step 5.2.2.2

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

$a = \pm 0$

Step 5.2.2.3

Plus or minus $0$ is $0$ .

$a = 0$

Step 6

Set $a^{2} - 2 a - 1$ equal to $0$ and solve for $a$ .

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Step 6.1

Set $a^{2} - 2 a - 1$ equal to $0$ .

$a^{2} - 2 a - 1 = 0$

Step 6.2

Solve $a^{2} - 2 a - 1 = 0$ for $a$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $a$ .

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

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

$a = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$a = \frac{2 \pm \sqrt{4 - 4 \cdot - 1}}{2 \cdot 1}$

Step 6.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

$a = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 6.2.3.1.3

Add $4$ and $4$ .

$a = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.2.3.1.4.1

Factor $4$ out of $8$ .

$a = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 6.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$a = \frac{2 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 6.2.3.1.5

Pull terms out from under the radical.

$a = \frac{2 \pm 2 \sqrt{2}}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$a = \frac{2 \pm 2 \sqrt{2}}{2}$

Step 6.2.3.3

Simplify $\frac{2 \pm 2 \sqrt{2}}{2}$ .

$a = 1 \pm \sqrt{2}$

Step 6.2.4

The final answer is the combination of both solutions.

$a = 1 + \sqrt{2}, 1 - \sqrt{2}$

Step 7

The final solution is all the values that make $- 4 a^{2} (a^{2} - 2 a - 1) = 0$ true.

$a = 0, 1 + \sqrt{2}, 1 - \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = 0, 1 + \sqrt{2}, 1 - \sqrt{2}$

Decimal Form:

$a = 0, 2.41421356 \dots, - 0.41421356 \dots$