Solve for x 5^(3x^2+3)=25^(12x-21)

$5^{3 x^{2} + 3} = 25^{12 x - 21}$

Step 1

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

$5^{3 x^{2} + 3} = 5^{2 (12 x - 21)}$

Step 2

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

$3 x^{2} + 3 = 2 (12 x - 21)$

Step 3

Solve for $x$ .

Simplify $2 (12 x - 21)$ .

Rewrite.

$3 x^{2} + 3 = 0 + 0 + 2 (12 x - 21)$

Simplify by adding zeros.

$3 x^{2} + 3 = 2 (12 x - 21)$

Apply the distributive property.

$3 x^{2} + 3 = 2 (12 x) + 2 \cdot - 21$

Multiply.

Multiply $12$ by $2$ .

$3 x^{2} + 3 = 24 x + 2 \cdot - 21$

Multiply $2$ by $- 21$ .

$3 x^{2} + 3 = 24 x - 42$

Subtract $24 x$ from both sides of the equation.

$3 x^{2} + 3 - 24 x = - 42$

Add $42$ to both sides of the equation.

$3 x^{2} + 3 - 24 x + 42 = 0$

Add $3$ and $42$ .

$3 x^{2} - 24 x + 45 = 0$

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} - 24 x + 45$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) - 24 x + 45 = 0$

Factor $3$ out of $- 24 x$ .

$3 (x^{2}) + 3 (- 8 x) + 45 = 0$

Factor $3$ out of $45$ .

$3 x^{2} + 3 (- 8 x) + 3 \cdot 15 = 0$

Factor $3$ out of $3 x^{2} + 3 (- 8 x)$ .

$3 (x^{2} - 8 x) + 3 \cdot 15 = 0$

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 15$ .

$3 (x^{2} - 8 x + 15) = 0$

Factor.

Factor $x^{2} - 8 x + 15$ using the AC method.

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$3 ((x - 5) (x - 3)) = 0$

Remove unnecessary parentheses.

$3 (x - 5) (x - 3) = 0$

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

$x - 5 = 0$

$x - 3 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $3 (x - 5) (x - 3) = 0$ true.

$x = 5, 3$

Solve for x 5^(3x^2+3)=25^(12x-21)

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