If a > 1.5 is a real number that satisfies the equation 4^a - 3(2^{a+1}) + 8 = 0, what is … — ACT

If $a > 1.5$ is a real number that satisfies the equation $4^a - 3(2^{a+1}) + 8 = 0$ , what is the value of $a$ ?

Answer: 2

Answer

The correct answer is 2.

Substituting

u = 2^a

turns the equation into

u^2 - 6u + 8 = 0

. Factoring gives

(u-2)(u-4)=0

, meaning

u=2

u=4

. Reversing the substitution gives

2^a = 2 \implies a=1

and

2^a = 4 \implies a=2

. Since

a > 1.5

, the correct value is 2.

Step-by-Step Solution

Express the equation in terms of base 2.

(2^a)^2 - 6(2^a) + 8 = 0

Since 4 is

2^2

and

2^{a+1} = 2 \cdot 2^a

, expressing all terms in base 2 allows for algebraic substitution.

Substitute

u = 2^a

to form a quadratic equation.

u^2 - 6u + 8 = 0

Substitution simplifies the exponential equation into a standard quadratic form.

Factor the quadratic equation.

(u-2)(u-4) = 0

Factoring allows us to find the roots of the quadratic equation.

Solve for the variable

a

a = 1

a = 2

Solving

2^a = 2

yields

a = 1

, and solving

2^a = 4

yields

a = 2

Apply the given constraint on

a

a = 2

The problem states that

a > 1.5

, so

a = 1

is discarded and

a = 2

is the correct value.

Key Concept

Solving exponential equations using quadratic substitution

Alternative Method

Instead of using substitution, test values for

a

. Since

a > 1.5

, testing small integer values starting with

a=2

shows

4^2 - 3(2^3) + 8 = 16 - 24 + 8 = 0

, validating that

a=2

is the solution.

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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