If 22x+4 – 17 × 2x+1 = –4, then which of the following is true?

Concept: An exponential equation whose exponents are linear in x (for example 22x+4 and 2x+1) can be reduced to a polynomial equation by substituting y for the repeating base raised to the smaller exponent. Every term is then rewritten as a power of y, the resulting polynomial is solved for y, and each valid value of y is converted back to x using the definition of the exponent.

Working:

1. Let y = 2x+1.

2. Since 2x+4 = 2(x+1)+2, write 22x+4 as 4 × (2x+1)2, i.e. 4y2.

3. Substitute into the equation: 4y2 − 17y = −4, i.e. 4y2 − 17y + 4 = 0.

4. Solve the quadratic using the quadratic formula: y = (17 ± √(172 − 4×4×4)) / (2×4) = (17 ± √(289−64)) / 8 = (17 ± 15) / 8.

5. This gives two values: y = 4 or y = 1/4.

6. Back-substitute y = 4: 2x+1 = 4 = 22, so x + 1 = 2, so x = 1, a positive value.

7. Back-substitute y = 1/4: 2x+1 = 1/4 = 2−2, so x + 1 = −2, so x = −3, a negative value.

Cross-check: Substitute x = 1 back into the original equation: 26 − 17 × 22 = 64 − 68 = −4, which matches. Substitute x = −3: 2−2 − 17 × 2−2 = 0.25 − 4.25 = −4, which also matches. Both roots satisfy the equation.

Since the equation has one positive root (x = 1) and one negative root (x = −3), x can take either a positive value or a negative value.