Algebra Examples

$5^{(x + 1) (x - 2)} = 625$

Step 1

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

$5^{(x + 1) (x - 2)} = 5^{4}$

Step 2

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

$(x + 1) (x - 2) = 4$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $(x + 1) (x - 2)$ .

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Step 3.1.1

Expand $(x + 1) (x - 2)$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$x (x - 2) + 1 (x - 2) = 4$

Step 3.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 + 1 (x - 2) = 4$

Step 3.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 + 1 x + 1 \cdot - 2 = 4$

Step 3.1.2

Simplify and combine like terms.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2.1.2

Move $- 2$ to the left of $x$ .

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

Step 3.1.2.1.3

Multiply $x$ by $1$ .

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

Step 3.1.2.1.4

Multiply $- 2$ by $1$ .

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

Step 3.1.2.2

Add $- 2 x$ and $x$ .

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

Step 3.2

Subtract $4$ from both sides of the equation.

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

Step 3.3

Subtract $4$ from $- 2$ .

$x^{2} - x - 6 = 0$

Step 3.4

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 3.4.2

Write the factored form using these integers.

$(x - 3) (x + 2) = 0$

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 $0$ .

$x - 3 = 0$

$x + 2 = 0$

Step 3.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.6.2

Add $3$ to both sides of the equation.

$x = 3$

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 $(x - 3) (x + 2) = 0$ true.

$x = 3, - 2$