Precalculus Examples

$x^{2} \cdot 10^{x} - 4 x \cdot 10^{x} = 32 (10^{x})$

Step 1

Subtract $32 \cdot 10^{x}$ from both sides of the equation.

$x^{2} \cdot 10^{x} - 4 x \cdot 10^{x} - 32 \cdot 10^{x} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $10^{x}$ out of $x^{2} \cdot 10^{x} - 4 x \cdot 10^{x} - 32 \cdot 10^{x}$ .

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Step 2.1.1

Factor $10^{x}$ out of $x^{2} \cdot 10^{x}$ .

$10^{x} \cdot x^{2} - 4 x \cdot 10^{x} - 32 \cdot 10^{x} = 0$

Step 2.1.2

Factor $10^{x}$ out of $- 4 x \cdot 10^{x}$ .

$10^{x} \cdot x^{2} + 10^{x} \cdot (- 4 x) - 32 \cdot 10^{x} = 0$

Step 2.1.3

Factor $10^{x}$ out of $- 32 \cdot 10^{x}$ .

$10^{x} \cdot x^{2} + 10^{x} \cdot (- 4 x) + 10^{x} \cdot - 32 = 0$

Step 2.1.4

Factor $10^{x}$ out of $10^{x} \cdot x^{2} + 10^{x} \cdot (- 4 x)$ .

$10^{x} \cdot (x^{2} - 4 x) + 10^{x} \cdot - 32 = 0$

Step 2.1.5

Factor $10^{x}$ out of $10^{x} \cdot (x^{2} - 4 x) + 10^{x} \cdot - 32$ .

$10^{x} (x^{2} - 4 x - 32) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{2} - 4 x - 32$ using the AC method.

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Step 2.2.1.1

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 $- 32$ and whose sum is $- 4$ .

$- 8, 4$

Step 2.2.1.2

Write the factored form using these integers.

$10^{x} ((x - 8) (x + 4)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$10^{x} (x - 8) (x + 4) = 0$

Step 3

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

$10^{x} = 0$

$x - 8 = 0$

$x + 4 = 0$

Step 4

Set $10^{x}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $10^{x}$ equal to $0$ .

$10^{x} = 0$

Step 4.2

Solve $10^{x} = 0$ for $x$ .

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Step 4.2.1

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{x}) = \log (0)$

Step 4.2.2

The equation cannot be solved because $\log (0)$ is undefined.

Undefined

Step 4.2.3

There is no solution for $10^{x} = 0$

No solution

Step 5

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

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Step 5.1

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

$x - 8 = 0$

Step 5.2

Add $8$ to both sides of the equation.

$x = 8$

Step 6

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

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Step 6.1

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

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The final solution is all the values that make $10^{x} (x - 8) (x + 4) = 0$ true.

$x = 8, - 4$