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

Polynomial, Quadratic and Root Relations short notes for CAT: 30 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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}}

The CAT Method: Convert Everything to S and P

Use the $S, P$ engine

Let $S = α + β$ and $P = α β$ . Most CAT root questions become simple after this naming.

From the equation, write

S = - \frac{b}{a}

and

P = \frac{c}{a}

Rewrite the asked expression using

S

and

P

Only solve individual roots if the expression cannot be handled by

S

and

P

Memory hook: For CAT quadratics, roots are not people. You rarely need their names; you need their team stats: sum and product.

Pattern 1: Reciprocal Roots

Reciprocal root questions

PYQ Pattern

If roots are $α, β$ , reciprocal expressions are usually just $S, P$ in disguise.

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

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

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

α^{2} + β^{2} = S^{2} - 2 P

Exam trigger: If you see “reciprocals of roots”, immediately write

S = α + β

and

P = α β

Pattern 2: One Common Root

Common root method

If multiple equations share a root, do not solve each equation fully.

Equation 1

r

satisfies it

Equation 2

r

satisfies it

Standard steps:

Let the common root be $r$ .
Substitute $x = r$ into each equation.
Subtract equations to eliminate $r^{2}$ .
Use the remaining relation to find the required expression.

This pattern is very common when coefficients contain unknowns like

m, n, p, q

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"]

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

Polynomial, Quadratic and Root Relations short notes for CAT: 30 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice quest

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

The Core Formula Card

Vieta Relations

The CAT Method: Convert Everything to S and P

Use the S,P engine

Pattern 1: Reciprocal Roots

Reciprocal root questions

Pattern 2: One Common Root

Common root method

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

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Use the $S, P$ engine