Linear Quadratic Answers

Quadratic Equations Problems : CAT 2007 Quant / Math Prep

Question 4 the day : March 12, 2003
The question for the day is from the topic Quadratic Equations. The question is about finding the value of an expression after evaluating the roots of a quadratic equation.

Question
If p and q are the roots of the equation x² - bx + c = 0, then what is the equation if the roots are (pq + p + q) and (pq - p - q)?

(1) x² - 2cx + (c² - b²) = 0
(2) x² - 2bx + (b² + c²) = 0
(3) Bcx² - 2(b+c)x + c² = 0
(4) x² + 2bx - (c² - b²) = 0

Correct Choice is (1). Correct Answer is x² - 2cx + (c² - b²) = 0

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Explanatory Answer

In the given quadratic equation x² - bx + c = 0,
The sum of the roots p + q = b --- (1)
And the product of the roots pq = c --- (2)

We have to formulate a quadratic equation whose roots are (pq + p + q) and (pq - p - q).

The sum of the two roots = pq + p +q + pq - p - q = 2pq
But from eqn (2), we know that pq = c
Therefore, the sum of the roots = 2c

The product of the roots = (pq + p + q)(pq - p - q)= (pq)² - (p+q)²
From equation (1) and (2), we know that pq = c and p + q = b
Therefore, the product of the roots = c² - b²

We know the sum of the roots and the product of the roots.
Therefore, the quadratic equation is x² - (sum of the roots)x + product of the roots = 0

=> x² - 2cx + c² - b² = 0

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