MathematicsChapter 2: Quadratic Equations

Equations That Reduce To Quadratic

Sometimes a substitution makes a quadratic: e.g.

Learning objectives

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.

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 inner quadratic, then go back to x. Reject values that make a denominator zero.

Mathematical form

u = 4 or u = 1

x = +/- 2 or x = +/- 1

u = x² u² - 5u + 4

u = 4 or u

Where / symbols

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 x^2: If you meet x^4 - 5x^2 + 4 = 0, put u = x^2. Then u^2 - 5u + 4 = 0, (u-4)(u-1)=0, u=4 or 1, so x^2=4 or x^2=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.

Overview

Hide the mess, then unwrap:

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.

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.

Examples x != values that zero a bottom · Extraneous roots possible · Substitute back

WHAT — EQUATIONS THAT REDUCE TO QUADRATIC

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 = x² 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.

Worked example

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

In simple words

u = x^2 cannot be negative. If a quadratic in u gave u=-3, that branch would give no real x.

Key points

substitute, solve, return
check domain
u = x²
LCD then quadratic

Exam preparation

Exam Focus

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