Solve for x in the equation, 9^x+1 + 32x = 270

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11. Solve for $x$ in the equation, $9^{x+1} + 3^{2x} = 270$

Express all terms with the same base.
Rewrite $9^{x+1}$ as $(3^2)^{x+1}$ .
$(3^2)^{x+1} + 3^{2x} = 270$ $3^{2(x+1)} + 3^{2x} = 270$ $3^{2x+2} + 3^{2x} = 270$
Factor out the common term $3^{2x}$ .
$3^{2x} \cdot 3^2 + 3^{2x} = 270$ $3^{2x}(3^2 + 1) = 270$ $3^{2x}(9 + 1) = 270$ $3^{2x}(10) = 270$
Isolate the exponential term.
$3^{2x} = \frac{270}{10}$ $3^{2x} = 27$
Express the right side with the same base as the left side.
Since $27 = 3^3$ :
$3^{2x} = 3^3$
Equate the exponents and solve for $x$ .
$2x = 3$ $x = \frac{3}{2}$
The value of $x$ is $\boxed{\frac{3}{2}}$ .

For the next question, the image shows:
Given $y = x^2 - 2x + 6$ , find the...
The rest of the question is cut off. Assuming the question asks for the minimum value of $y$ or the coordinates of the vertex of the parabola:
Identify the coefficients of the quadratic equation.
The equation is in the form $y = ax^2 + bx + c$ , where $a=1$ , $b=-2$ , and $c=6$ .
Since $a > 0$ , the parabola opens upwards, meaning it has a minimum value at its vertex.
Find the x-coordinate of the vertex.
The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$ .
$x = -\frac{-2}{2(1)}$ $x = \frac{2}{2}$ $x = 1$
Find the minimum value of $y$ (the y-coordinate of the vertex).
Substitute the x-coordinate of the vertex ( $x=1$ ) back into the equation for $y$ .
$y = (1)^2 - 2(1) + 6$ $y = 1 - 2 + 6$ $y = 5$
The minimum value of $y$ is $\boxed{5}$ .
The coordinates of the vertex are $\boxed{(1, 5)}$ .

Solve for x in the equation, 9^x+1 + 32x = 270

Answer

Step-by-step solution

Express all terms with the same base.

Factor out the common term $3^{2x}$ .

Isolate the exponential term.

Express the right side with the same base as the left side.

Equate the exponents and solve for $x$ .

Identify the coefficients of the quadratic equation.

Find the x-coordinate of the vertex.

Find the minimum value of $y$ (the y-coordinate of the vertex).

Try one yourself

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More Mathematics Questions

Solve for x in the equation, 9^x+1 + 32x = 270

Answer

Step-by-step solution

Express all terms with the same base.

Factor out the common term 32x3^{2x}32x.

Isolate the exponential term.

Express the right side with the same base as the left side.

Equate the exponents and solve for xxx.

Identify the coefficients of the quadratic equation.

Find the x-coordinate of the vertex.

Find the minimum value of yyy (the y-coordinate of the vertex).

Try one yourself

Still stuck on this one?

More Mathematics Questions

Factor out the common term $3^{2x}$ .

Equate the exponents and solve for $x$ .

Find the minimum value of $y$ (the y-coordinate of the vertex).