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Equations That Reduce to Quadratic
Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inner quadratic, then go back to x. Reject values that make a denominator zero.
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Equations That Reduce to Quadratic
What is it?
Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inner quadratic, then go back to x. Reject values that make a denominator zero.
In simple words
- Sometimes a substitution makes a quadratic: e.g.
- IN x2: If you meet x4 - 5x2 + 4 = 0, put u = x2.
- AFTER CLEARING FRACTIONS: A rational equation can become quadratic.
- Worked-method reminder: x = +/- 2 or x = +/- 1
Important terms
EQUATIONS THAT REDUCE TO QUADRATIC: Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inner quadratic, then go back to x. Reject values that make a denominator zero.
IN x2: If you meet x4 - 5x2 + 4 = 0, put u = x2. Then u2 - 5u + 4 = 0, (u-4)(u-1)=0, u=4 or 1, so x2=4 or x2=1, x= +/-2 or +/-1.
AFTER CLEARING FRACTIONS: A rational equation can become quadratic. Multiply by the LCD, then it is Chapter 5 plus this chapter. Always list excluded x first.
What you should learn
- Explain equations that reduce to quadratic using the correct definitions, conditions, and notation.
- Apply substitute, solve, return, check domain correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- u=4 -> x=+/-2
- u=1 -> x=+/-1
- Four real roots here
IN x^2
If you meet x4 - 5x2 + 4 = 0, put u = x2. Then u2 - 5u + 4 = 0, (u-4)(u-1)=0, u=4 or 1, so x2=4 or x2=1, x= +/-2 or +/-1.
AFTER CLEARING FRACTIONS
A rational equation can become quadratic. Multiply by the LCD, then it is Chapter 5 plus this chapter. Always list excluded x first.
Recommended for Class 10
Full school-level explanation
Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inner quadratic, then go back to x. Reject values that make a denominator zero.
Hide the mess, then unwrap:
WHAT — EQUATIONS THAT REDUCE TO QUADRATIC
Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inner quadratic, then go back to x. Reject values that make a denominator zero. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Equations That Reduce To Quadratic matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through IN x^2, then AFTER CLEARING FRACTIONS. Worked: solve x^4 - 5x^2 + 4 = 0. let u = x^2 u^2 - 5u + 4 = 0 (u - 4)(u - 1) = 0 u = 4 or u = 1 x^2 = 4 or x^2 = 1 x = +/- 2 or x = +/- 1 Finish by checking signs, restrictions, units, and whether the answer matches the question.
IN x^2
If you meet x4 - 5x2 + 4 = 0, put u = x2. Then u2 - 5u + 4 = 0, (u-4)(u-1)=0, u=4 or 1, so x2=4 or x2=1, x= +/-2 or +/-1.
Remember
- u = x2
Examples
- u=4 -> x=+/-2
- u=1 -> x=+/-1
- Four real roots here
AFTER CLEARING FRACTIONS
A rational equation can become quadratic. Multiply by the LCD, then it is Chapter 5 plus this chapter. Always list excluded x first.
Remember
- LCD then quadratic
Examples
- x != values that zero a bottom
- Extraneous roots possible
- Substitute back
Remember
- substitute, solve, return
- check domain
- x = +/-2 or +/-1
- u = x2 >= 0
- u = x2
- LCD then quadratic
- substitute, solve, return
- check domain
How the ideas connect
Worked: solve x4 - 5x2 + 4 = 0. let u = x2 u2 - 5u + 4 = 0 (u - 4)(u - 1) = 0 u = 4 or u = 1 x2 = 4 or x2 = 1 x = +/- 2 or x = +/- 1
Deeper understanding
Go beyond memorising Equations That Reduce to Quadratic. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve x4 - 5x2 + 4 = 0. let u = x2 u2 - 5u + 4 = 0 (u - 4)(u - 1) = 0 u = 4 or u = 1 x2 = 4 or x2 = 1 x = +/- 2 or x = +/- 1
Common mistakes
Exam focus
Core Idea
IN x^2
AFTER CLEARING FRACTIONS
Worked Method
Exam focus
Quick revision
- substitute, solve, return
- check domain
- x = +/-2 or +/-1
- u = x2 >= 0
- EQUATIONS THAT REDUCE TO QUADRATIC: Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a f
- IN x2: If you meet x4 - 5x2 + 4 = 0, put u = x2. Then u2 - 5u + 4 = 0, (u-4)(u-1)=0, u=4 or 1, so x2=4
u = x^2 cannot be negative. If a quadratic in u gave u=-3, that branch would give no real x.
For long answers on Equations That Reduce to Quadratic: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- substitute, solve, return
- check domain
- x = +/-2 or +/-1
- u = x2 >= 0
Important formulas
- substitute, solve, return
- check domain
- x = +/-2 or +/-1
Common exam traps
- u = x^2 cannot be negative. If a quadratic in u gave u=-3, that branch would give no real x.
- Do not treat IN x^2 and AFTER CLEARING FRACTIONS as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
u = x^2 cannot be negative. If a quadratic in u gave u=-3, that branch would give no real x.
- Mistake
Do not treat IN x^2 and AFTER CLEARING FRACTIONS as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
u = x^2 cannot be negative. If a quadratic in u gave u=-3, that branch would give no real x.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
substitute, solve, return
check domain
x = +/-2 or +/-1
u = x2 >= 0
Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inn
x = +/-2 or +/-1
- Definition
- Sometimes a substitution makes a quadratic: e.g. an equation in x^4 that is quadratic in x^2, or a fraction that becomes quadratic after clearing. Solve the inn
- Formula
- x = +/-2 or +/-1
- Example
- u=4 -> x=+/-2
- Critical warning
- u = x^2 cannot be negative. If a quadratic in u gave u=-3, that branch would give no real x.
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Exam Practice
Define
Q1. Define Equations That Reduce to Quadratic.
Explain
Q2. Explain: IN x^2.
Example
Q3. Give one clear example of Equations That Reduce to Quadratic and say why it fits.
Mistake
Q4. State one common mistake students make about Equations That Reduce to Quadratic.
Exam note
Q5. Write a short exam note on Equations That Reduce to Quadratic: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.