Solve for x 4^(x^2-11x+30)=16

$4^{x^{2} - 11 x + 30} = 16$

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

$4^{x^{2} - 11 x + 30} = 4^{2}$

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

$x^{2} - 11 x + 30 = 2$

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$x^{2} - 11 x + 30 - 2 = 0$

Subtract $2$ from $30$ .

$x^{2} - 11 x + 28 = 0$

Factor $x^{2} - 11 x + 28$ 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 $28$ and whose sum is $- 11$ .

$- 7, - 4$

Write the factored form using these integers.

$(x - 7) (x - 4) = 0$

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

$x - 7 = 0$

$x - 4 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x - 7) (x - 4) = 0$ true.

$x = 7, 4$

Solve for x 4^(x^2-11x+30)=16

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