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Determine the type of the two roots of each of the following equations:
- (1) \( x^2 - 3x + 5 = 0 \)
- (2) \( x^2 + 10x + 25 = 0 \)
- (3) \( 3x^2 + 10x = 4 \)
Solution
(1) \( \because a = 1, \, b = -3, \, c = 5 \)
\( \therefore \) Discriminant \( = (-3)^2 - 4 \times 1 \times 5 = -11 \) (negative)
\( \therefore \) The roots are complex and non‑real.
(2) \( \because a = 1, \, b = 10, \, c = 25 \)
\( \therefore \) Discriminant \( = 100 - 100 = 0 \)
\( \therefore \) The roots are real and equal.
(3) \( \because a = 3, \, b = 10, \, c = -4 \)
\( \therefore \) Discriminant \( = 100 - 4 \times 3 \times (-4) = 100 + 48 = 148 \) (positive)
\( \therefore \) The roots are real and distinct.
Created with Mr. Ayman Hassan for math egypt lovers