Concept 10 / 10Complex Numbers and Quadratic Equations
Common Roots, Equations Reducible to Quadratics and Maximum-Minimum of Rational Expressions
Conditions for two quadratics to share one or both roots, solving equations that become quadratic after substitution (reciprocal, biquadratic, exponential, modulus), and extreme values of quadratic and rational expressions.
Common roots. Suppose and share a root . Treat and as two unknowns in the two linear equations and . By cross-multiplication,
.
Since , the condition for a common root is
,
and the common root itself is . In practice it is often faster to subtract suitable multiples of the two equations to eliminate , solve the resulting linear equation for , and substitute back.
Both roots common: the two quadratics must be proportional, . A useful special case: if two quadratics with real coefficients share one non-real root, they share both (since non-real roots come in conjugate pairs), so they are proportional. Similarly, if the coefficients are rational and they share one irrational root , they share both.
Equations reducible to quadratics. The idea is to spot a substitution that turns the equation into a quadratic in , solve for , then solve for each valid , rejecting values of outside the range of .
Biquadratic, : set .
Reciprocal equations, such as (coefficients symmetric): divide by and set , using . For real , .
Products of four linear factors, with : pair them as , and both brackets share the part , which becomes the substitution.
Exponential, : set .
Modulus, such as : set , using .
Radical, such as : square, but then check each solution in the original equation, since squaring can introduce extra roots (here we need ).
Extreme values of expressions. For , the extreme value is at the vertex; on a closed interval, compare the vertex value (if the vertex is inside the interval) with the endpoint values. For , rewrite as a quadratic in with coefficients depending on , and require a real root, so its discriminant must be . This gives an inequality in whose solution set is the range. Remember to handle the value of at which the coefficient vanishes, and to check that the obtained does not make the denominator zero.
Key Takeaways:
- One common root: .
- Both roots common: .
- Real-coefficient quadratics sharing a non-real root share both roots.
- Substitutions: , (), (), ().
- After substituting, reject values of outside the range of the substitution.
- After squaring, verify every candidate in the original equation.
- Range of quadratic over quadratic: discriminant in must be .
- Pitfall: accepting in , or in when real is required.
Brain-Teaser 1: Find such that and have a common root.
Solution: Subtract the equations: , that is, . If , the equations are identical () and share both roots, and . Otherwise the common root is , and substituting into the second equation gives , so . Check: with the equations are and , which share the root . So (one common root) or (both common, non-real). Subtracting the equations to find the common root is much faster than the general formula.
Brain-Teaser 2: Solve .
Solution: Pair the factors so the sums match: . Then and . Let : , so , giving or . For : , so or . For : has , giving the non-real roots . The real solutions are and . Check: . The pairing trick turns a quartic into two quadratics.
Brain-Teaser 3: Solve .
Solution: Write and . With : , so or . Both are positive, so both are valid: gives , and gives . If one of the -values had been negative or zero it would have to be rejected, since for all real . The substitution turns an exponential equation into a quadratic, and the range of the substitution is the filter for extraneous values.
Practice this concept
Free · no account required · instant solutions