Algebra Examples

$2^{x^{2} + x + 0.5} = 4 \sqrt{2}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$2^{x^{2} + x + 0.5} = 4 \cdot 2^{\frac{1}{2}}$

Step 2

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

$2^{x^{2} + x + 0.5} = 2^{2} \cdot 2^{\frac{1}{2}}$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{x^{2} + x + 0.5} = 2^{2 + \frac{1}{2}}$

Step 4

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2^{x^{2} + x + 0.5} = 2^{2 \cdot \frac{2}{2} + \frac{1}{2}}$

Step 5

Combine $2$ and $\frac{2}{2}$ .

$2^{x^{2} + x + 0.5} = 2^{\frac{2 \cdot 2}{2} + \frac{1}{2}}$

Step 6

Combine the numerators over the common denominator.

$2^{x^{2} + x + 0.5} = 2^{\frac{2 \cdot 2 + 1}{2}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $2$ by $2$ .

$2^{x^{2} + x + 0.5} = 2^{\frac{4 + 1}{2}}$

Step 7.2

Add $4$ and $1$ .

$2^{x^{2} + x + 0.5} = 2^{\frac{5}{2}}$

Step 8

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x^{2} + x + 0.5 = \frac{5}{2}$

Step 9

Solve for $x$ .

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Step 9.1

Subtract $\frac{5}{2}$ from both sides of the equation.

$x^{2} + x + 0.5 - \frac{5}{2} = 0$

Step 9.2

Simplify $x^{2} + x + 0.5 - \frac{5}{2}$ .

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Step 9.2.1

To write $0.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + x + 0.5 \cdot \frac{2}{2} - \frac{5}{2} = 0$

Step 9.2.2

Combine $0.5$ and $\frac{2}{2}$ .

$x^{2} + x + \frac{0.5 \cdot 2}{2} - \frac{5}{2} = 0$

Step 9.2.3

Combine the numerators over the common denominator.

$x^{2} + x + \frac{0.5 \cdot 2 - 5}{2} = 0$

Step 9.2.4

Simplify the numerator.

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Step 9.2.4.1

Multiply $0.5$ by $2$ .

$x^{2} + x + \frac{1 - 5}{2} = 0$

Step 9.2.4.2

Subtract $5$ from $1$ .

$x^{2} + x + \frac{- 4}{2} = 0$

Step 9.2.5

Divide $- 4$ by $2$ .

$x^{2} + x - 2 = 0$

Step 9.3

Factor $x^{2} + x - 2$ using the AC method.

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Step 9.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 9.3.2

Write the factored form using these integers.

$(x - 1) (x + 2) = 0$

Step 9.4

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

$x - 1 = 0$

$x + 2 = 0$

Step 9.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 9.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 9.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9.6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 9.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 9.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9.7

The final solution is all the values that make $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$