Quadratic Equations
Integral roots of quadratic
nta_pyq_2025_apr
Grade 12
Question:
Consider the equation$x + 4x - n = 0$, where n$\i_n$[20, 100] is a natural number. Then the number of all distinct 2 values of n, for which the given equation has integral roots, is equal to
Step-by-Step Solution
Key Concept: Rewrite as$(x+2)^2=n+4$and require integer roots by making$n+4$a perfect square.
$x + 4x + 4 = n + 4$2 2$(3)$($x + 2) = n + 4$x = -2 $\pm$$\sqrtn + 4$∵ 20$\le n$$\le 100$$\sqrt{2}4$$\le$$\sqrtn + 4$$\le$$\sqrt{10}4$$\Rightarrow$$\sqrtn + 4$$\i_n {5, 6, 7, 8, 9, 10}$∴ ′ 6 ' integral values of ' n ' are possible
Correct Answer: 3