Algebra Examples

$7^{x^{2}} = 49^{x + 4}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$7^{x^{2}} = 7^{2 (x + 4)}$

Step 2

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

$x^{2} = 2 (x + 4)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (x + 4)$ .

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Step 3.1.1

Apply the distributive property.

$x^{2} = 2 x + 2 \cdot 4$

Step 3.1.2

Multiply $2$ by $4$ .

$x^{2} = 2 x + 8$

Step 3.2

Subtract $2 x$ from both sides of the equation.

$x^{2} - 2 x = 8$

Step 3.3

Subtract $8$ from both sides of the equation.

$x^{2} - 2 x - 8 = 0$

Step 3.4

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 3.4.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.4.2

Write the factored form using these integers.

$(x - 4) (x + 2) = 0$

Step 3.5

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

$x - 4 = 0$

$x + 2 = 0$

Step 3.6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.7

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

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Step 3.7.1

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

$x + 2 = 0$

Step 3.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.8

The final solution is all the values that make $(x - 4) (x + 2) = 0$ true.

$x = 4, - 2$