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Worked solution

Use a positive exponential substitution

Solve the equation 4x+1=2−7(2x)4^{x+1}=2-7(2^x).

CJC · O Level revision · Example 8(b)

Answer

x=−2x=-2.

Compare your reasoning

Rewrite the left side in base 2

(22)x+1=2−7(2x) ⟹ 22x+2=2−7(2x) ⟹ 4(22x)+7(2x)−2=0\begin{aligned}&(2^2)^{x+1}\\ & =2-7(2^x)\ \Longrightarrow\ 2^{2x+2}\\ & =2-7(2^x)\ \Longrightarrow\ 4(2^{2x})\\ &{}+7(2^x)-2\\ & =0\end{aligned}

The source writes 4=2², making the squared exponential visible.

Solve the quadratic in a positive variable

4y2+7y−2=0 ⟹ (4y−1)(y+2)=0 ⟹ y=14 or y=−2\begin{aligned}&4y^2+7y-2\\ & =0\ \Longrightarrow\ (4y-1)(y+2)\\ & =0\ \Longrightarrow\ y\\ & =\frac14\ \text{or}\ y\\ & =-2\end{aligned}

Let y=2xy=2^x. Its positivity is essential when selecting a quadratic root.

Reject the negative candidate and solve

2x=14=2−2 ⟹ x=−22^x=\frac14=2^{-2}\ \Longrightarrow\ x=-2

The source explicitly rejects 2x=−22^x=-2 because 2x>02^x>0.

How did this question go?

Solution notes

This is printed Example 8(b). For Example 8(a), use Tutorial Question 9, which asks the same equation.