Reducing an equation to quadratic form
Some equations contain the same expression in several places. Give that whole expression a temporary name, solve the resulting quadratic in the temporary variable and then reverse the substitution
\[u=\text{repeated expression}\]
- Name the repeated expression Replace the complete repeated base, including all its terms, with u \(\displaystyle u=2x-1\)
- Solve the outer quadratic Solve in u before returning to the original variable \(\displaystyle u^2-5u+6=0\Rightarrow(u-2)(u-3)=0\)
- Substitute back Use each temporary value in the substitution and solve for x \(\displaystyle 2x-1=2\Rightarrow x=\frac{3}{2}\)
- Check restrictions Reject temporary or original values that cannot satisfy the substitution or the original equation \(\displaystyle u=x^2\Rightarrow u\ge0\)
Worked example
Solve
\[(x+1)^2-7(x+1)+12=0\]
- \(\displaystyle u=x+1\)Replace the repeated expression x + 1 with u
- \(\displaystyle u^2-7u+12=0\Rightarrow(u-3)(u-4)=0\)Solve the resulting quadratic by factorising
- \(\displaystyle u=3\text{ or }u=4\)Apply the null factor law to obtain the temporary values
- \(\displaystyle x+1=3\text{ or }x+1=4\Rightarrow x=2\text{ or }x=3\)Substitute each temporary value back and solve for x
Common mistakes
Not reversing the substitution
✕\(\displaystyle u=5\Rightarrow x=5\)✓\(\displaystyle u=5\Rightarrow x^2+4=5\Rightarrow x=\pm1\)
A root found for u is not an x-value until it is substituted back into the definition of uKeeping an invalid temporary value
✕\(\displaystyle u=x^2,\quad u=-2\Rightarrow x=\pm\sqrt{-2}\)✓\(\displaystyle u=x^2,\quad u=-2\text{ is rejected because }u\ge0\)
A square cannot be negative, so a negative value of u is outside the substitution's domainSkipping the original-equation check
✕\(\displaystyle \sqrt{x+1}=x-1\Rightarrow x=0\text{ or }x=3\)✓\(\displaystyle \sqrt{x+1}=x-1\Rightarrow x=3\)
Squaring can introduce an extraneous candidate, so each result must be tested in the original equationDifficulty levels
introUse a repeated linear expression to reduce an equation to a factorised quadratic
standardSolve the temporary quadratic and reverse the substitution for all x-values
challengeHandle a repeated nonlinear expression and reject values outside its domain
examReduce, solve, back-substitute and check every candidate in the original equation
Questions about this topic
What should be substituted for u?+
Use the complete expression that repeats in the equation, including its brackets and all terms
Why must the temporary values be substituted back?+
The temporary equation gives values of u, not directly values of x. The definition of u is needed to find the corresponding x-values
Why can a value be rejected after solving?+
The substitution may restrict the possible values, or algebra such as squaring may create a candidate that does not satisfy the original equation
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations
In this chapter
Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems