Finite Math Examples

$7^{x^{2} - 96} = 49^{2 x}$

Step 1

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

$7^{x^{2} - 96} = 7^{2 (2 x)}$

Step 2

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

$x^{2} - 96 = 2 (2 x)$

Step 3

Solve for $x$ .

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Step 3.1

Multiply $2$ by $2$ .

$x^{2} - 96 = 4 x$

Step 3.2

Subtract $4 x$ from both sides of the equation.

$x^{2} - 96 - 4 x = 0$

Step 3.3

Factor $x^{2} - 96 - 4 x$ using the AC method.

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Step 3.3.1

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 $- 96$ and whose sum is $- 4$ .

$- 12, 8$

Step 3.3.2

Write the factored form using these integers.

$(x - 12) (x + 8) = 0$

Step 3.4

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

$x - 12 = 0$

$x + 8 = 0$

Step 3.5

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 3.5.2

Add $12$ to both sides of the equation.

$x = 12$

Step 3.6

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 3.6.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3.7

The final solution is all the values that make $(x - 12) (x + 8) = 0$ true.

$x = 12, - 8$