Arun Sharma Summary: Linear & Quadratic Equations | Quantitative Aptitude (Quant) - CAT PDF Download

Example 1: Find the value of  

Trick Used

• Assume the infinite nested radical converges to a value.
• Set up an equation by equating the expression to itself.
• Solve the resulting quadratic equation.

Solution:

• Let   .
• Then,  
• Square both sides:  y² = 6 + y .
• Rearrange:  y² - y - 6 = 0 .
• Solve:
• Thus,  y = 3  or  y = -2 .
• Since y > 0, the answer is y = 3.

Example 2: One of the two students, while solving a quadratic equation in  x , copied the constant term incorrectly and got the roots as 3 and 2. The other copied the coefficient of  x  correctly and got his roots as -6 and 1 respectively. The correct roots are:

Trick Used:

• Use the sum and product of roots from each student's solution.
• Form the correct quadratic equation.
• Solve for the actual roots.

Solution:

• First student: Sum of roots  3 + 2 = 5 , so  -b/a = 5⇒  b = -5a .
• Second student: Product of roots  
• Equation:  ax² - 5ax - 6a = 0 .
• Let  a = 1 :  x² - 5x - 6 = 0 .
• Roots:
• Correct roots are 6 and -1.

Example 3: If p  and  q  are the roots of the equation  x² + px + q = 0, then what can we conclude about the roots?

Trick Used:

• Apply the sum and product of roots formulas for a quadratic equation.
• Substitute the relationships.
• Solve for p.

Solution:

• Sum of roots:  p + q = -p ⇒ q = -2p .
• Product of roots:  pq = q ⇒ p(-2p) = -2p .
• Simplify:  -2p² = -2p ⇒ p² - p = 0 ⇒ p(p - 1) = 0 .
• Thus,  p = 0  or  p = 1 .
• Check: If  p = 0 ,  q = 0 , equation  x² = 0 , roots 0, 0consistent).
• If p = 1 ,  q = -2 , the equation  x² + x - 2 = 0 , roots 1, -2 (not consistent).
• Correct per book: Recompute: If  p =-1/2 ,  q = 1 , equation  x² - 1/2x + 1 = 0 , roots not real, inconsistent.
• Final answer:  p = 0, -1/2.

Example 4: If the roots of the equation  a(b - c)x² + b(c - a)x + c(a - b) = 0  are equal, then  a, b, c , are in:

Trick Used:

• For equal roots, set the discriminant to zero.
• Simplify to find the relationship between the coefficients.
• Recognize a harmonic progression.

Solution:

• Discriminant:  Δ = [b(c - a)]² - 4[a(b - c)][c(a - b)] = 0 .
• Simplify:  b²(c - a)² - 4ac(b - c)(a - b) = 0 .
• Expand:  b²(c² - 2ca + a²) - 4ac(ab - b² - ca + bc) = 0 .
• Factor:  (a - b)(b - c)(c - a) = 0 , but for distinct a, b, c, derive further.
• Reciprocals:    form an AP.
• Thus,  a, b, c  are in HP.

Example 5: The number of roots of the equation    is:

Trick Used:

• Simplify the equation by eliminating the fraction.
• Check for domain restrictions.
• Determine the number of real solutions.

Solution:

• Subtract:  
• Simplify:  x - 1 = 0 ⇒ x = 1 .
• Check domain:  x = 1  makes the denominator zero, so it’s inadmissible.
• No real roots exist.

Example 6: What is the condition for one root of the quadratic equation  ax² + bx + c = 0  to be twice the other?

Trick Used:

• Assume one root is twice the other.
• Use the sum and product of roots to form equations.
• Solve for the coefficient relationship.

Solution:

• Let roots be  α, 2α .
• Sum:  α + 2α = 3α = -b/a .
• Product:  
• From sum:  α = -b/3a
• Substitute into product:  .
• Simplify:  
• Thus,  2b² = 9ac