Nonlinear equations and mixed systems
| English | 中文 | Pinyin |
|---|---|---|
| intersection/ˌɪntəˈsekʃn/ | 交点 | jiāo diǎn |
| extraneous root/ekˈstreɪnɪəs ruːt/ | 增根 | zēng gēn |
A decision before an answer
- Squaring √(x+2)=x creates candidate roots; it does not guarantee that a negative right side can equal a square root.
- Your goal: Solve quadratic, absolute-value, rational and radical equations.
Read the relationship
- Use structure before expanding. A factored quadratic equals zero when at least one factor is zero. An absolute-value equation |u|=a has no real solution for a<0, one condition u=0 when a=0 and two conditions u=±a when a>0. Account for all allowed cases, not just the positive branch.
- Reject candidate roots violating the original equation or domain.
Which solves √(x+6)=x with real x?
Squaring gives x=3 or -2; only 3 makes both original sides nonnegative and equal.
Use the defining rule
- For a rational equation, exclude denominator zeros before multiplying through. Multiplication may create a candidate at an excluded value. For a radical equation, note that a principal square root is nonnegative, isolate it and square carefully. Squaring is a one-way implication until every resulting candidate is checked in the original.
- Find intersections of a line and a nonlinear curve.
How many real solutions has |x+1|=-2?
Absolute value cannot be negative.
Check the conditions
- To solve a line–parabola system, substitute the line expression into the quadratic. The real roots give x-values; use the line to recover each y. No real root means no real intersection. A repeated quadratic root gives tangency, while two different real roots give two intersections.
- Find intersections of a line and a nonlinear curve.
√(x+2)=x requires x≥0. Squaring yields x²-x-2=0, so x=2 or -1; only 2 works in the original. For y=x+2 and y=x², solve x²-x-2=0: x=2,-1, giving (2,4) and (-1,1). For |2x-1|=5, solve 2x-1=5 or -5 to obtain x=3 or -2.
For (x-5)(x+1)=0, the negative solution is ____.
The second factor is zero at -1.
Apply the task format
- A graph or calculator can help locate candidates and check scale, but rounded intersections are not exact proof when the question asks an exact value or solution count. Relate graphical behaviour to the algebraic equation. A double root should not be counted twice as two different ordered pairs.
- Find intersections of a line and a nonlinear curve.
Do not report every root of a transformed equation before checking the original restriction and branch.
Which answer fits this case?
Solve quadratic, absolute-value, rational and radical equations
A repeated root gives two distinct intersections.
Multiplicity does not create distinct ordered pairs.
Keep the distinctions
- extraneous root 增根 — A transformed-equation solution that fails the original equation.
- intersection 交点 — An ordered pair satisfying both curve equations.
- Solve quadratic, absolute-value, rational and radical equations.
- Reject candidate roots violating the original equation or domain.
- Find intersections of a line and a nonlinear curve.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.