Look for one expression and its square. Put u equal to that expression, solve the quadratic in u, and then solve for the original variable. Record the possible values of u before accepting a quadratic root. For x4/3-5x2/3+4=0, put u=x2/3=([3]x)2 0. Then (u-1)(u-4)=0. Thus [3]x=+-1,+-2, giving x=+-1,+-8. A fractional even power can produce both signs of x. For exponential equations, u= ex>0. For logarithmic equations, require positive logarithm arguments first; the logarithm itself can have either sign. Always reverse the substitution.

Look for one expression and its square. Put \(u\) equal to that expression, solve the quadratic in \(u\), and then solve for the original variable. Record the possible values of \(u\) before accepting a quadratic root.

For \(x^{4/3}-5x^{2/3}+4=0\), put \(u=x^{2/3}=(\sqrt[3]x)^2\ge0\). Then \((u-1)(u-4)=0\). Thus \(\sqrt[3]x=\pm1,\pm2\), giving \(x=\pm1,\pm8\). A fractional even power can produce both signs of \(x\).

For exponential equations, \(u=\mathrm e^x>0\). For logarithmic equations, require positive logarithm arguments first; the logarithm itself can have either sign. Always reverse the substitution.

Recognising a repeated expression

Recognising a repeated expression