CUET UG Mathematics: Algebra - Quadratic Equations, Nature of Roots, Sum/Product of Roots

Must-Know

• A quadratic equation is of the form ( ax^2 + bx + c = 0 ), where ( a \neq 0 ), and ( a, b, c \in \mathbb{R} ). Example: ( 2x^2 - 5x + 3 = 0 ).
• The discriminant ( D ) of a quadratic equation ( ax^2 + bx + c = 0 ) is given by ( D = b^2 - 4ac ). Example: For ( x^2 - 4x + 4 = 0 ), ( D = (-4)^2 - 4(1)(4) = 0 ).
• If ( D > 0 ), roots are real and distinct. Example: ( x^2 - 5x + 6 = 0 ) has ( D = 25 - 24 = 1 > 0 ), roots 2 and 3.
• If ( D = 0 ), roots are real and equal. Example: ( x^2 - 6x + 9 = 0 ), ( D = 36 - 36 = 0 ), root 3 (repeated).
• If ( D < 0 ), roots are complex and conjugate. Example: ( x^2 + 2x + 5 = 0 ), ( D = 4 - 20 = -16 < 0 ), roots ( -1 \pm 2i ).
• The sum of roots ( \alpha + \beta = -\frac{b}{a} ). Example: For ( 3x^2 - 7x + 2 = 0 ), sum = ( \frac{7}{3} ).
• The product of roots ( \alpha\beta = \frac{c}{a} ). Example: For ( 4x^2 - 8x + 3 = 0 ), product = ( \frac{3}{4} ).
• If roots are ( \alpha ) and ( \beta ), the quadratic equation is ( x^2 - (\alpha + \beta)x + \alpha\beta = 0 ). Example: Roots 2 and -3-equation ( x^2 - (-1)x + (-6) = x^2 + x - 6 = 0 ).
• For a quadratic equation with rational coefficients, irrational roots occur in conjugate pairs. Example: If ( 2 + \sqrt{3} ) is a root, so is ( 2 - \sqrt{3} ).
• For a quadratic equation with real coefficients, complex roots occur as conjugate pairs. Example: If ( 1 + i ) is a root, so is ( 1 - i ).
• If one root of ( 2x^2 - kx + 3 = 0 ) is 1, then substituting gives ( 2(1)^2 - k(1) + 3 = 0 \Rightarrow k = 5 ).
• The nature of roots can be determined without solving the equation using the discriminant. Example: ( x^2 + x + 1 = 0 ), ( D = 1 - 4 = -3 < 0 ), so complex roots.
• A quadratic equation has equal roots if and only if ( D = 0 ). Example: ( 9x^2 - 6x + 1 = 0 ), ( D = 36 - 36 = 0 ), root ( \frac{1}{3} ).
• If the roots of ( x^2 - px + q = 0 ) are reciprocal, then ( \alpha \cdot \frac{1}{\alpha} = 1 = q \Rightarrow q = 1 ). Example: ( x^2 - 5x + 1 = 0 ), roots reciprocal.
• If one root is zero, then ( c = 0 ). Example: ( 2x^2 - 3x = 0 ), roots 0 and ( \frac{3}{2} ).
• If both roots are zero, then ( b = 0 ) and ( c = 0 ), but ( a \ne 0 ). Example: ( 4x^2 = 0 ), root 0 (double).
• The minimum or maximum value of ( ax^2 + bx + c ) occurs at ( x = -\frac{b}{2a} ). Example: ( -x^2 + 4x + 1 ) has maximum at ( x = 2 ).
• If ( \alpha, \beta ) are roots of ( x^2 - 3x + 2 = 0 ), then ( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 9 - 4 = 5 ).
• The condition for the roots to be equal in magnitude but opposite in sign is sum = 0? ( -\frac{b}{a} = 0 \Rightarrow b = 0 ). Example: ( x^2 - 4 = 0 ), roots 2 and -2.
• The condition for roots to be reciprocal is product = 1? ( \frac{c}{a} = 1 \Rightarrow c = a ). Example: ( 3x^2 - 10x + 3 = 0 ), roots 3 and ( \frac{1}{3} ).

Difficulty Level

Intermediate — requires understanding of discriminant, root relationships, and algebraic manipulation, but no advanced theorems.

Common CUET Traps

• Trap: Assuming that if discriminant is negative, roots are "not real" but forgetting they are complex conjugates.
  Avoid: Always write complex roots as conjugate pairs when coefficients are real.
• Trap: Using sum and product formulas with incorrect signs (e.g., sum = ( \frac{b}{a} ) instead of ( -\frac{b}{a} )).
  Avoid: Remember: sum = ( -\text{coefficient of }x / \text{coefficient of }x^2 ).
• Trap: Assuming irrational roots like ( \sqrt{2} ) can occur alone in equations with rational coefficients.
  Avoid: Irrational roots occur in conjugate pairs only if coefficients are rational.

Practice MCQs

1. Question: What is the nature of roots of ( 2x^2 - 4x + 3 = 0 )?
   A) Real and equal
   B) Real and distinct
   C) Complex
   D) Rational
   Answer: C
   Explanation: ( D = (-4)^2 - 4(2)(3) = 16 - 24 = -8 < 0 ), so roots are complex.
   Why others fail: Option B is tempting if student miscalculates discriminant as positive.

2. Question: If the sum of roots of a quadratic equation is 5 and product is 6, what is the equation?
   A) ( x^2 + 5x + 6 = 0 )
   B) ( x^2 - 5x + 6 = 0 )
   C) ( x^2 - 6x + 5 = 0 )
   D) ( x^2 + 6x + 5 = 0 )
   Answer: B
   Explanation: Equation is ( x^2 - (\text{sum})x + \text{product} = x^2 - 5x + 6 = 0 ).
   Why others fail: Option A is tempting if student forgets sign of sum.

3. Question: If one root of ( 3x^2 - 9x + k = 0 ) is 2, what is the value of ( k )?
   A) 3
   B) 6
   C) 9
   D) 12
   Answer: B
   Explanation: Substituting ( x = 2 ): ( 3(4) - 9(2) + k = 0 \Rightarrow 12 - 18 + k = 0 \Rightarrow k = 6 ).
   Why others fail: Option C is tempting if student miscalculates ( 3(4) = 12 ) but adds instead of subtracts.

4. Question: For what value of ( p ) does the equation ( x^2 - 2px + 16 = 0 ) have equal roots?
   A) 2
   B) 4
   C) 8
   D) 16
   Answer: B
   Explanation: For equal roots, ( D = 0 \Rightarrow (-2p)^2 - 4(1)(16) = 0 \Rightarrow 4p^2 = 64 \Rightarrow p^2 = 16 \Rightarrow p = \pm 4 ), so ( p = 4 ) is valid.
   Why others fail: Option C is tempting if student sets ( 2p = 16 ) directly.

5. Question: If the roots of ( x^2 - 7x + 12 = 0 ) are ( \alpha ) and ( \beta ), what is ( \alpha^2 + \beta^2 )?
   A) 13
   B) 24
   C) 25
   D) 49
   Answer: C
   Explanation: ( \alpha + \beta = 7 ), ( \alpha\beta = 12 ), so ( \alpha^2 + \beta^2 = 7^2 - 2(12) = 49 - 24 = 25 ).
   Why others fail: Option D is tempting if student forgets to subtract ( 2\alpha\beta ).

Last-Minute Revision

• Discriminant ( D = b^2 - 4ac ) determines nature of roots.
• ( D > 0 ): real and distinct; ( D = 0 ): real and equal; ( D < 0 ): complex conjugates.
• Sum of roots = ( -\frac{b}{a} ), not ( \frac{b}{a} ).
• Product of roots = ( \frac{c}{a} ).
• Equation from roots: ( x^2 - (\text{sum})x + \text{product} = 0 ).
• If roots are reciprocal, product = 1? ( c = a ).
• If roots are equal and opposite, sum = 0? ( b = 0 ).
• If one root is zero, constant term ( c = 0 ).
• Complex roots occur in conjugate pairs for real coefficients.
• Irrational roots (e.g., ( a + \sqrt{b} )) occur in conjugate pairs if coefficients are rational.
• For equal roots, ( D = 0 ) is necessary and sufficient.
• Minimum/maximum of quadratic occurs at ( x = -\frac{b}{2a} ).
• ( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta ).
• ( (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta ).
• If roots are equal, vertex lies on x-axis.
• Mnemonic: "Sum = -b/a, Product = c/a"-think "SP" = "Same as c/a, Negative b/a".
• Verify from NCERT: exact wording of nature of roots in Chapter 4, Class 10 NCERT.
• Verify from NCERT: derivation of sum and product formulas in Class 10, Chapter 4.