Question

Match the number and nature of the solutions to each equation.

3x2 – 5x + 3 = 0
25x2 – 10x = –1
x2 + 5x + 3 = 0
3x2 + 5x = –2

Answer 1

Use the discriminant D = b^2 − 4ac.

1) 3x^2 − 5x + 3 = 0
D = (−5)^2 − 4·3·3 = 25 − 36 = −11 < 0 → no real solutions (two complex conjugates).

2) 25x^2 − 10x = −1 → 25x^2 − 10x + 1 = 0
D = (−10)^2 − 4·25·1 = 100 − 100 = 0 → one real double root: x = 1/5.

3) x^2 + 5x + 3 = 0
D = 5^2 − 4·1·3 = 25 − 12 = 13 > 0 → two distinct real irrational roots: x = [−5 ± √13]/2.

4) 3x^2 + 5x = −2 → 3x^2 + 5x + 2 = 0
D = 5^2 − 4·3·2 = 25 − 24 = 1 > 0 → two distinct real rational roots: x = −1 and x = −2/3.