Polynomial, Quadratic and Root Relations Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Polynomial, Quadratic and Root Relations notes for CAT: 59 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Polynomial, Quadratic and Root Relations

Chapter journey

Polynomial, Quadratic and Root Relations

🌱 Topic 1: Quadratic Roots and Vieta Relations

You learn to convert roots into two numbers: sum and product. This is the selected topic.

CAT PYQ count in this topic: 5 | Importance: moderate but core

🔍 Topic 2: Discriminant, Integer Roots and Root Conditions

You decide what type of roots exist using conditions like real roots, no real roots, equal roots, and integer roots.

CAT PYQ count in this topic: 4

🧩 Topic 3: Polynomial Roots and Conjugate Surds

You extend root logic to polynomial expressions and special root pairs such as conjugate surds.

CAT PYQ count in this topic: 5

By the end of this chapter, the goal is simple: do not fear roots. You should know when to solve, when to use sum-product, and when to use root conditions.

Topic Hero: Roots Without Solving

Selected Topic

Quadratic Roots and Vieta Relations

CAT often hides the roots. Your job is to extract what matters without wasting time.

Main conversion

If roots are

α, β

a x^{2} + b x + c = 0

then

α + β = - \frac{b}{a}, α β = \frac{c}{a}

See roots?

Think sum and product.

See reciprocals?

Divide by product.

See common root?

Let it be $r$ .

Why Vieta Works: Coefficients Remember the Roots

Coefficients secretly store root data

If roots are

α

and

β

, then a quadratic can be written as:

a (x - α) (x - β) = 0

Expanding:

a (x - α) (x - β) = a [x^{2} - (α + β) x + α β]

So compared with:

a x^{2} + b x + c = 0

we get:

α + β = - \frac{b}{a}, α β = \frac{c}{a}

CAT lesson: when a question asks about roots indirectly, first write their sum and product. Solving the roots is usually slower.

The Core Formula Card

Must know

Vieta Relations

Summary

Quadratic	Roots	Sum	Product
$a x^{2} + b x + c = 0$	$α, β$	$α + β = - \frac{b}{a}$	$α β = \frac{c}{a}$

α^{2} + β^{2} = (α + β)^{2} - 2 α β

α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)

\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}

\frac{1}{α ^{2}} + \frac{1}{β ^{2}} = \frac{α ^{2} + β ^{2}}{α ^{2} β ^{2}}

Polynomial, Quadratic and Root Relations: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability NAT

The equations $x^{2} - 7 x + p = 0$ and $x^{2} - 3 x + q = 0$ have exactly one common root. If $p + q = 12$ , then the number of possible values of the sum of the other roots is

Correct Answer:

Step-by-Step Solution

Key idea: this is a sum-of-other-roots common-root question, recognisable because two quadratics share one root and the question asks about the roots that are not common.

Step 1: Let the common root be $r$ .

Step 2: For $x^{2} - 7 x + p = 0$ , the sum of roots is $7$ . If one root is $r$ , the other root is $7 - r$ . Therefore

$p = r (7 - r) .$

Step 3: For $x^{2} - 3 x + q = 0$ , the sum of roots is $3$ . If one root is $r$ , the other root is $3 - r$ . Therefore

$q = r (3 - r) .$

Step 4: Use $p + q = 12$ :

$r (7 - r) + r (3 - r) = 12.$

Step 5: Simplify:

$10 r - 2 r^{2} = 12.$

Step 6: Rearrange:

$2 r^{2} - 10 r + 12 = 0.$

Step 7: Divide by $2$ :

$r^{2} - 5 r + 6 = 0.$

Step 8: Factor:

$(r - 2) (r - 3) = 0.$

Hence $r = 2$ or $r = 3$ .

Step 9: The sum of the other roots is

$(7 - r) + (3 - r) = 10 - 2 r .$

Step 10: Check the two cases.

If $r = 2$ , the sum is $10 - 4 = 6$ .

If $r = 3$ , the sum is $10 - 6 = 4$ .

These are two distinct possible values.

Answer: 2

Question 2 · Quantitative Ability MSQ

Let $f (x) = a x^{2} + b x + c$ where $a, b, c$ are real numbers and $a \neq = 0$ . Which of the following statements is ALWAYS true if the discriminant $Δ = b^{2} - 4 a c$ is negative?

I. The expression $a f (x)$ is positive for all real $x$ .

II. The equation $f (x) = k$ has no real solutions for any real $k$ .

III. The values $f (0)$ and $f (1)$ have the same sign.

A.
I only
B.
I and III only
C.
II and III only
D.
I, II and III

Correct Answer:

["B"]

Step-by-Step Solution

Key idea: Conceptual understanding of $Δ < 0$ . Negative discriminant means no real roots, so $f (x)$ never crosses zero. Thus $f (x)$ maintains a constant sign identical to $a$ .

Analysis of Statement I:

Since $Δ < 0$ , $f (x)$ has the same sign as $a$ for all $x$ .

Therefore, $a \cdot f (x) = a \cdot (same sign as a) = a^{2} \cdot (positive factor) > 0$ .

More simply: if $a > 0, f > 0 ⟹ a f > 0$ . If $a < 0, f < 0 ⟹ a f > 0$ .

Statement I is ALWAYS TRUE.

Analysis of Statement II:

$f (x) = k ⟺ a x^{2} + b x + (c - k) = 0$ .

Discriminant of this new equation: $D_{k} = b^{2} - 4 a (c - k) = b^{2} - 4 a c + 4 ak = Δ + 4 ak$ .

We know $Δ < 0$ . Can we choose $k$ such that $D_{k} \geq 0$ ?

Yes. If $a > 0$ , choose large positive $k$ . If $a < 0$ , choose large negative $k$ .

Geometrically: A parabola that doesn't touch x-axis still covers a range of y-values. Any $k$ in that range yields solutions.

Statement II is FALSE.

Analysis of Statement III:

Since $f (x)$ never changes sign (continuous function with no zeros), $f (x)$ is either always positive or always negative.

Therefore, $f (0)$ and $f (1)$ must have the same sign.

Statement III is ALWAYS TRUE.

Conclusion: I and III are true. Option B corresponds to "I and III only".

Answer: ["B"]

More notes in this unit

chapter

Polynomial, Quadratic and Root Relations Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Polynomial, Quadratic and Root Relations notes for CAT: 59 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

The equations x^{2} - 7 x + p = 0 and x^{2} - 3 x + q = 0 have exactly one common root. If p + q = 12, then the number of possible values of the sum of the other roots is

Question 2

Let f (x) = a x^{2} + b x + c where a, b, c are real numbers and a \neq = 0 . Which of the following statements is ALWAYS true if the discriminant Δ = b^{2} - 4 a c is negative? I. The expression a f (x) is positive for all real x . II. The equation f (x) = k has no real solutions for any real k . III. The values f (0) and f (1) have the same sign.

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