Solve for x 8^(x^2)=2^(2x+1)

$8^{x^{2}} = 2^{2 x + 1}$

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

$2^{3 x^{2}} = 2^{2 x + 1}$

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

$3 x^{2} = 2 x + 1$

Solve for $x$ .

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

$3 x^{2} - 2 x = 1$

Subtract $1$ from both sides of the equation.

$3 x^{2} - 2 x - 1 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

Factor $- 2$ out of $- 2 x$ .

$3 x^{2} - 2 x - 1 = 0$

Rewrite $- 2$ as $1$ plus $- 3$

$3 x^{2} + (1 - 3) x - 1 = 0$

Apply the distributive property.

$3 x^{2} + 1 x - 3 x - 1 = 0$

Multiply $x$ by $1$ .

$3 x^{2} + x - 3 x - 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + x) - 3 x - 1 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (3 x + 1) - (3 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$(3 x + 1) (x - 1) = 0$

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

$3 x + 1 = 0$

$x - 1 = 0$

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(3 x + 1) (x - 1) = 0$ true.

$x = - \frac{1}{3}, 1$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{3}, 1$

Decimal Form:

$x = - 0.\overline{3}, 1$

Solve for x 8^(x^2)=2^(2x+1)

Related Questions

Support LCMGCF.com by disabling your adblocker.