Equations quadratic in another expression
| English | 中文 | Pinyin |
|---|---|---|
| substitution/ˌsʌbstɪˈtjuːʃn/ | 代换 | dài huàn |
A fourth-degree equation can hide an ordinary quadratic. Solving for the new variable is only half the job.
- A fourth-degree equation can hide an ordinary quadratic. Solving for the new variable is only half the job.
- This lesson studies substitution 代换: Replacing a repeated expression with a new variable to reveal a simpler equation.
Choose the mathematical structure
- Identify a repeated expression and write u for it, including any restrictions. Solve the resulting quadratic in u. Then solve the original expression equal to each allowed u value. Substitute every final candidate into the original equation; a solution for u is not yet a solution for x.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines substitution?
Replacing a repeated expression with a new variable to reveal a simpler equation.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x⁴−5x²+4=0, set u=x² with u≥0. Then u²−5u+4=(u−1)(u−4)=0, so x²=1 or 4 and x=±1,±2. For x⁴+x²−2=0, the u-equation is (u−1)(u+2)=0: u=−2 is impossible for real x, leaving x=±1. For (x+1)²−5(x+1)+6=0, set u=x+1: u=2 or 3 gives x=1 or 2. For 3^(2x)−4×3^x+3=0, set u=3^x>0: u=1 or 3 gives x=0 or 1.
Equations quadratic in another expression
Identify a repeated expression and write u for it, including any restrictions
Find the first valid or invalid step in a quadratic method.
Find the largest real solution of x⁴−5x²+4=0.
The u roots 1 and 4 give x=±1,±2; the largest is 2.
Test a tempting shortcut
- Do not report u=1 and u=4 as the x solutions of the quartic. Taking a square root needs both signs, but a principal square-root expression is nonnegative. A quadratic in u can have two real roots while one or both violate u’s domain.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Two roots in the substituted variable always give exactly two real solutions in x. This claim is false. Explain which definition or assumption it violates.
Find the smaller solution of (x+1)²−5(x+1)+6=0.
u=x+1 gives u=2 or 3, hence x=1 or 2.
Two roots in the substituted variable always give exactly two real solutions in x.
Do not report u=1 and u=4 as the x solutions of the quartic. Taking a square root needs both signs, but a principal square-root expression is nonnegative. A quadratic in u can have two real roots while one or both violate u’s domain.
Interpret a new situation
- Choose a substitution that repeats exactly: 3^(2x)=(3^x)², while 3^(x²) is a different expression. Count distinct real x solutions after returning to x, not before. Check by direct substitution: x=−2 gives 16−20+4=0 in the first quartic.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many distinct real solutions does x⁴−5x²+4=0 have?
The four distinct values −2,−1,1,2 all satisfy the original equation.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Replacing a repeated expression with a new variable to reveal a simpler equation. Choose the relationship, show the method, check its assumptions and interpret the result.