Quadratic Equations Test

Result

Quadratic Equations Test - 1

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Question 1
1 / -0

The equation ax² + bx + c = 0 has equal roots. Based on this information, which of the following is correct?

A
b² – 4ac = 0

B
b² – 4ac > 0

C
b² – 4ac < 0

D
b² – 4ac ≥ 0

Solution

General rules:
If b² – 4ac = 0, then there are two equal real roots.
If b² – 4ac > 0, then there are two distinct real roots.
If b² – 4ac < 0, then there are no real roots.
Question 2
1 / -0

How many positive integers will 'k' take, such that the roots of the quadratic equation 2x² - 13x + k = 0 are real?

A
13

B
21

C
22

D
8

Solution

Since the roots are real, therefore b² - 4ac ≥ 169 - 8k ≥ 0.
8k ≤ 169
k ≤ 21.1
So, 'k' will take 21 positive integers (from 1 to 21).
Question 3
1 / -0

If 3 is a real root of the equation x² + bx – 15 = 0 and equation x² + bx + q = 0 has equal roots, then what is the value of q?

A
1

B
-1

C
2

D
-2

Solution

x = 3 is a real root of the equation, so:
x² + bx – 15 = 0
3² + b × 3 – 15 = 0
9 + 3b – 15 = 0
3b – 6 = 0
3b = 6
b = 2
Now, the equation x² + bx + q = 0 becomes x² + 2x + q = 0 --- (I)
Since eq (I) has equal roots:
b² – 4ac = 0
2² – 4 × 1 × q = 0
4 – 4q = 0
-4q = -4
q = 1