Exponential equations reducible to quadratics — 9709 P3
Solve a quadratic in a positive exponential variable.
Notes and key methods
When an equation contains \(\displaystyle a^{2x}\) and \(\displaystyle a^{x}\), use \(\displaystyle u=a^{x}\) with u>0. Solve the quadratic in u, reject nonpositive u-values, and recover x using powers or logarithms.
Distinct positive u-values give distinct real x-values because \(\displaystyle a^{x}\) is one-to-one when a>0 and \(\displaystyle a\ne 1\). Check the original exponential equation after recovery.
Check the conditions
The substitution \(\displaystyle u=a^{x}\) requires u>0.
The base is positive and different from one.
Three worked examples
Example 1 · Exponential equations via substitution
Solve \(2^{2x}-34(2^x)+64=0\).
Solve the equation, showing the substitution.
Find the larger solution x.
Find the product of the two x-solutions.
Find the difference between the larger and smaller x-solutions.
Show the reasoning
(a)
Put U=2ˣ>0. Then \(\left(U - 32\right) \left(U - 2\right)\)=0; the two positive powers give the stated exponents.
Answer: \(\left\{1, 5\right\}\)
(b)
Recover x from the second positive power.
\(5\)
Answer: \(5\).
(c)
Multiply the recovered exponents, not the substituted roots.
\(5\)
Answer: \(5\).
(d)
Subtract the two exponents.
\(4\)
Answer: \(4\).
Example 2 · Exponential equations via substitution
Solve \(3^{2x}-252(3^x)+2187=0\).
Solve the equation, showing the substitution.
Find the larger solution x.
Find the product of the two x-solutions.
Find the difference between the larger and smaller x-solutions.
Show the reasoning
(a)
Put U=3ˣ>0. Then \(\left(U - 243\right) \left(U - 9\right)\)=0; the two positive powers give the stated exponents.
Answer: \(\left\{2, 5\right\}\)
(b)
Recover x from the second positive power.
\(5\)
Answer: \(5\).
(c)
Multiply the recovered exponents, not the substituted roots.
\(10\)
Answer: \(10\).
(d)
Subtract the two exponents.
\(3\)
Answer: \(3\).
Example 3 · Exponential equations via substitution
Solve \(4^{2x}-16448(4^x)+1048576=0\).
Solve the equation, showing the substitution.
Find the larger solution x.
Find the product of the two x-solutions.
Find the difference between the larger and smaller x-solutions.
Show the reasoning
(a)
Put U=4ˣ>0. Then \(\left(U - 16384\right) \left(U - 64\right)\)=0; the two positive powers give the stated exponents.
Answer: \(\left\{3, 7\right\}\)
(b)
Recover x from the second positive power.
\(7\)
Answer: \(7\).
(c)
Multiply the recovered exponents, not the substituted roots.
\(21\)
Answer: \(21\).
(d)
Subtract the two exponents.
\(4\)
Answer: \(4\).
Common mistakes
A root in u is not yet an answer for x.
Do not accept a negative value for \(\displaystyle a^{x}\).