Comparing Roots of Quadratic Equations
Comparing Roots of Quadratic Equations involves determining which roots are greater, smaller or equal without unnecessary calculation. The main tools are the quadratic formula, discriminant, sum and product of roots, and careful comparison of radical expressions. This page explains the required formulas, sign rules, methods and examples for quantitative aptitude questions.
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What Is Comparing Roots of Quadratic Equations?
For an equation ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac))/(2a). The discriminant D = b² − 4ac must be non-negative for real-root comparison. If a > 0, the smaller root uses −√D and the larger root uses +√D; if a < 0, the order is reversed because the denominator is negative.
Comparing Roots of Quadratic Equations Video Lessons
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Quadratic Equations: Comparing Roots
Learn how to compare the roots of quadratic equations and determine which root is larger using algebraic relationships and sign-based reasoning.
Comparing Roots of Quadratic Equations Formula & Tricks
Important Formulas
For ax² + bx + c = 0, the two roots are obtained by using the plus and minus signs.
D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots.
These are Vieta's relations for roots α and β of ax² + bx + c = 0.
When a is negative, the two expressions exchange their order because division by a negative number reverses inequalities.
Quick Tricks
Multiplying or dividing every coefficient of a quadratic equation by the same non-zero number does not change its roots. Simplify the equation before comparing.
For non-negative A and B, A < B exactly when A² < B². This helps compare expressions containing square roots, but both sides must first be known to be non-negative.
When comparing fractions, a positive denominator preserves the inequality and a negative denominator reverses it. This is essential when the leading coefficient a is negative.
For f(x) = a(x − α)(x − β) with α < β, if a > 0 then f(x) is negative between the roots and positive outside them. Values at suitable test points can locate roots in intervals without calculating exact radicals.
Comparing Roots of Quadratic Equations Concepts
Checking Whether Real Roots Exist
For ax² + bx + c = 0, D = b² − 4ac. If D > 0, two distinct real roots exist; if D = 0, both roots are equal; if D < 0, the roots are non-real and ordinary real-number comparison is not applicable.
Finding the Smaller and Larger Root
If a > 0 and D ≥ 0, then α = (−b − √D)/(2a) and β = (−b + √D)/(2a), with α ≤ β. If a < 0, the denominator is negative, so these labels are reversed.
Comparing Roots from Two Equations
For two equations, first identify the smaller and larger root of each equation. Compare fractions by cross-multiplication only after checking denominator signs. For radical expressions, isolate the radicals and square only when both sides are non-negative.
Using Sum and Product of Roots
For ax² + bx + c = 0, α + β = −b/a and αβ = c/a. If αβ < 0, the roots have opposite signs. If αβ > 0 and α + β > 0, both roots are positive; if αβ > 0 and α + β < 0, both are negative, provided the roots are real.
Locating Roots Through Intervals
For f(x) = ax² + bx + c, if f(p) and f(q) have opposite signs, then a root lies between p and q. This method is especially useful when the quadratic is difficult to solve exactly. The complete ordering still requires checking both roots or using the sign pattern of the parabola.
Comparing Roots of Quadratic Equations Data Table
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Discriminant and Root Type
The discriminant determines the nature and equality of the roots.
| Condition on D | Nature of roots | Comparison conclusion |
|---|---|---|
| D > 0 | Two distinct real roots | The roots can be ordered and compared |
| D = 0 | Two equal real roots | Both roots are −b/(2a) |
| D < 0 | Two non-real roots | Real-number comparison is not applicable |
Note: Here D = b² − 4ac for ax² + bx + c = 0.
Root Signs from Sum and Product
The signs of α + β and αβ help determine the signs of real roots.
| Condition | Root sign information | Example pattern |
|---|---|---|
| αβ < 0 | One root positive and one root negative | x² − 3x − 4 = 0 |
| αβ > 0 and α + β > 0 | Both roots positive | x² − 5x + 6 = 0 |
| αβ > 0 and α + β < 0 | Both roots negative | x² + 5x + 6 = 0 |
| αβ = 0 | At least one root is zero | x² − 4x = 0 |
Note: These conclusions assume that the roots are real.
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Comparing Roots of Quadratic Equations: Revision Points
Use these rules to compare real roots accurately.
- For ax² + bx + c = 0, use D = b² − 4ac to check whether real roots exist.
- The roots are (−b ± √D)/(2a).
- When a > 0, the minus expression is the smaller root and the plus expression is the larger root.
- When a < 0, the order of the two quadratic-formula expressions is reversed.
- The sum and product of roots are −b/a and c/a, respectively.
- A negative product means that the two real roots have opposite signs.
- Do not determine individual root order from the sum and product alone.
- When squaring an inequality, first verify that both sides are non-negative.
Comparing Roots of Quadratic Equations FAQs
What is the formula for comparing roots of quadratic equations?
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac))/(2a). Calculate the ordered roots of each equation and compare their exact values.
How do you compare roots when the leading coefficient is negative?
The denominator 2a is negative, so the usual plus-minus order reverses. For example, the roots of −x² + 5x − 6 = 0 are 2 and 3, although the plus expression in the numerator produces the smaller value after division.
Can the sum and product of roots alone identify the larger root?
No. The sum and product identify the combined root information but generally do not specify which individual root is larger. The quadratic formula, factorisation or additional conditions are needed.
What happens when the discriminant is zero?
The quadratic has two equal real roots. Their common value is −b/(2a). For x² − 8x + 16 = 0, the repeated root is 4.
How can roots be compared without solving a quadratic completely?
Use the discriminant, sign of the quadratic at selected points, factorisation when available, and the sign pattern outside or between the roots. A sign change between p and q proves that a root lies in that interval.
What does a negative product of roots indicate?
Since αβ = c/a, αβ < 0 means the two real roots have opposite signs. One root is positive and the other is negative.