Consider the exponential equation: 4^x - 3 \cdot 2^{x+2} + 32 = 0 Determine the sum of all… — State PSC Exam

Consider the exponential equation:

4^x - 3 \cdot 2^{x+2} + 32 = 0

Determine the sum of all real values of $x$ that satisfy this equation.

Answer: 5

By applying the laws of indices, the original expression transforms into a quadratic equation in terms of

2^x

. Solving

y^2 - 12y + 32 = 0

yields

y=4

and

y=8

, which correspond precisely to

x=2

and

x=3

. Their sum is 5.

Express all terms with a common base of 2.

4^x

becomes

(2^x)^2

and

2^{x+2}

becomes

4 \cdot 2^x

Creating a common base allows the equation to be transformed into a standard polynomial form.

Rewrite the equation using the new terms.

(2^x)^2 - 12(2^x) + 32 = 0

Simplifying the coefficients makes it easier to spot the quadratic structure.

Perform a substitution to solve the quadratic equation.

Letting

y = 2^x

gives

y^2 - 12y + 32 = 0

. Factoring yields

(y - 4)(y - 8) = 0

, so

y = 4

y = 8

Substitution converts a complex exponential equation into a simple quadratic one.

Solve for the original variable

x

2^x = 4 \implies x = 2

, and

2^x = 8 \implies x = 3

The question asks for the values of

x

, not the intermediate substitution variable

y

Calculate the sum of all valid

x

values.

2 + 3 = 5

This addresses the specific final requirement of the question stem.

Solving exponential equations reducible to quadratics using index laws.

Estimated Time:1m 30s

Question