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Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT PDF Download

Importance of Quadratic Equations

In the CAT exam, quadratic equations are crucial as they frequently appear in the quantitative aptitude section. Proficiency in solving quadratic equations is essential for mastering algebraic concepts and ensuring success in the quantitative reasoning portion of the exam. A solid understanding of quadratic equations enables candidates to solve complex mathematical problems efficiently, showcasing their quantitative abilities to excel in the CAT exam.

Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT

Important Formulas

  • General Quadratic equation will be in the form of ax2 + bx + c = 0 The values of 'x' satisfying the equation are called roots of the equation.
  • The value of roots, p and q Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT
  • The above formula is known as the Shreedhara Acharya's Formula, after the ancient Indian Mathematician who derived it.
  • Sum of the roots = p + q = -b/a
  • Product of the roots = p × q = c/a
  • If 'c' and 'a' are equal then the roots are reciprocal to each other.
  • If b = 0, then the roots are equal and are opposite in sign.
  • Let D denote the discriminant, D = b2 - 4ac.
  • Depending on the sign and value of D, nature of the roots would be as follows:
  • D < 0 and |D| is not a perfect square: Roots will be in the form of p + iq and p - iq where p and q are the real and imaginary parts of the complex roots. p is rational and q is irrational.
  • D < 0 and |D| is a perfect square: Roots will be in the form of p + iq and p - iq where p and q are both rational.
  • D = 0 ⇒ Roots are real and equal. X = -b/2a
  • D > 0 and D is not a perfect square: Roots are conjugate surds
  • D > 0 and D is a perfect square: Roots are real, rational and unequal

Signs of the Roots

Let P be product of roots and S be their sum

  • P > 0, S > 0 : Both roots are positive
  • P > 0, S < 0 : Both roots are negative
  • P < 0, S > 0 : Numerical smaller root is negative and the other root is positive
  • P < 0, S < 0 : Numerical larger root is negative and the other root is positive
  • Minimum and maximum values of a2 + bx + c = 0
  • If a > 0: minimum value Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT and occurs at x = -b/2a
  • If a < 0: maximum value =Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT and occurs at =  -b/2a
  • Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT
  • Sum of the roots = Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT
  • Sum of roots taken two at a time = Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT
  • Sum of roots taken three at a time  Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT and so on Product of the roots Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT

Finding a Quadratic Equation

  • If roots are given:
    (x − a)(x − b) = 0 ⇒ x2 − (a + b)x + ab = 0
  • If sum s and product p of roots are given:
    x2 − sx + p = 0
  • If roots are reciprocals of roots of equation
    ax2 + bx + c = 0, then equation is
    cx2 + bx + a = 0
  • If roots are k more than roots of
    ax2 + bx + c = 0 then equation is
    a(x - k)2 + b(x - k) + c = 0
  • If roots are k more than roots of
    ax2 + bx + c = 0 then equation is
    a(x/k)2 + b(x/k) + c = 0
  • Descartes Rules: A polynomial equation with n sign changes can have a maximum of n positive roots. To find the maximum possible number of negative roots, find the number of positive roots of f(-x).
  • An equation where the highest power is odd must have at least one real root.
The document Important Formulas: Quadratic Equations | Quantitative Aptitude (Quant) - CAT is a part of the CAT Course Quantitative Aptitude (Quant).
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FAQs on Important Formulas: Quadratic Equations - Quantitative Aptitude (Quant) - CAT

1. What are the signs of the roots of a quadratic equation?
Ans. The signs of the roots of a quadratic equation can be determined by examining the coefficients of the equation. If the quadratic equation is of the form ax^2 + bx + c = 0, then the signs of the roots will be as follows: - If the discriminant (b^2 - 4ac) is positive, then the roots will be real and distinct. - If the discriminant is zero, then the roots will be real and equal. - If the discriminant is negative, then the roots will be complex conjugates with imaginary parts.
2. How can I find the quadratic equation given its roots?
Ans. If you know the roots of a quadratic equation, you can find the equation by using the fact that the sum and product of the roots can be related to the coefficients of the equation. Let the roots of the quadratic equation be r1 and r2. Then the quadratic equation can be written as (x - r1)(x - r2) = 0. Expanding this equation will give you the quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c can be determined using the coefficients of the expanded equation.
3. What are the important formulas for solving quadratic equations?
Ans. There are several important formulas for solving quadratic equations. Some of them include: - The quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. - The sum of roots formula: The sum of the roots of a quadratic equation -b/a. - The product of roots formula: The product of the roots of a quadratic equation c/a.
4. How can I determine if a quadratic equation has real or complex roots?
Ans. To determine if a quadratic equation has real or complex roots, you can calculate the discriminant, which is given by the expression b^2 - 4ac. If the discriminant is positive, then the roots will be real and distinct. If the discriminant is zero, then the roots will be real and equal. If the discriminant is negative, then the roots will be complex conjugates with imaginary parts.
5. Can a quadratic equation have only one real root?
Ans. Yes, a quadratic equation can have only one real root. This happens when the discriminant of the equation is zero. In this case, the quadratic equation will have a repeated root, meaning that both roots will be equal. The graph of the quadratic equation will touch the x-axis at only one point.
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