Chapter 4 — Quadratic Equations
Question Bank
A. MCQs — Choose the Correct Option
1.
Which of the following is in the standard form of a quadratic equation?
A. 3x+5=0
B. 2×2−7x+3=0
C. x3+x−1=0
D. x1+2=0
Answer: B
2.
In 5×2−3x+7=0, the values of a,b,c are:
A. 5,3,7
B. 5,−3,7
C. −5,3,7
D. 5,−3,−7
Answer: B
3.
Which condition must hold for ax2+bx+c=0 to be quadratic?
A. a=0
B. b=0
C. a=0
D. c=0
Answer: C
4.
The equation x(x+2)+3=2x+7
after simplification is:
A. x2+3=0
B. x2+2=0
C. x2−7=0
D. x+2=0
Answer: B
5.
The roots of (x−4)(x+3)=0
are:
A. 4,3
B. −4,3
C. 4,−3
D. −4,−3
Answer: C
6.
The discriminant of 2×2−5x+3=0
is:
A. 1
B. 25
C. 49
D. 13
Answer: A
7.
If the discriminant of a quadratic equation is negative, then it has:
A. two distinct real roots
B. two equal real roots
C. no real roots
D. exactly one positive root
Answer: C
8.
If b2−4ac=0, the roots are:
A. irrational
B. imaginary
C. distinct real numbers
D. equal real numbers
Answer: D
9.
The roots of x2−5x+6=0
are:
A. 1,6
B. 2,3
C. −2,−3
D. 3,5
Answer: B
10.
For 3×2+7x−2=0, the value of the discriminant is:
A. 25
B. 49
C. 73
D. 1
Answer: C
11.
Which method is especially useful when a quadratic expression can easily be written as a product of two linear factors?
A. Factorisation
B. Elimination
C. Substitution
D. Division
Answer: A
12.
The quadratic formula is:
A. 2ab±b2+4ac
B. 2a−b±b2−4ac
C. a−b±b2+4ac
D. ab±b2−4ac
Answer: B
13.
If a quadratic equation has two distinct real roots, its discriminant must be:
A. zero
B. negative
C. positive
D. undefined
Answer: C
14.
The equation x2+4x+4=0
has:
A. two distinct real roots
B. two equal real roots
C. no real roots
D. three roots
Answer: B
15.
The roots of x2−9=0 are:
A. 9,−9
B. 3,−3
C. 0,9
D. 3,9
Answer: B
B. Fill in the Blanks
1.
The standard form of a quadratic equation is __________.
Answer: ax2+bx+c=0, a=0
2.
The expression b2−4ac is called the __________.
Answer: discriminant
3.
A quadratic equation can have at most __________ roots.
Answer: two
4.
If b2−4ac>0, the equation has __________ distinct real roots.
Answer: two
5.
If b2−4ac=0, the roots are __________.
Answer: equal
6.
If b2−4ac<0, the equation has __________ real roots.
Answer: no
7.
A number that satisfies a quadratic equation is called a __________ of that equation.
Answer: root
8.
The roots of a quadratic equation are the same as the __________ of its corresponding quadratic polynomial.
Answer: zeroes
9.
The denominator in the quadratic formula is __________.
Answer: 2a
10.
For a quadratic equation, the coefficient of x2 cannot be __________.
Answer: zero
C. True or False
1.
Every equation containing x2 is necessarily a quadratic equation.
Answer: False
2.
A quadratic equation can have two equal real roots.
Answer: True
3.
A quadratic equation can have more than two real roots.
Answer: False
4.
The discriminant is b2−4ac.
Answer: True
5.
If the discriminant is negative, the quadratic has no real roots.
Answer: True
6.
Factorisation can be used to find roots when the quadratic expression can be factorised into linear factors.
Answer: True
7.
The coefficient a in ax2+bx+c=0 may be zero.
Answer: False
8.
An equation should always be simplified before deciding whether it is quadratic.
Answer: True
9.
A negative algebraic root must always be rejected.
Answer: False
Reason: It is rejected only when it is unsuitable for the original context, such as representing a physical length.
10.
When D=0, the quadratic has two distinct real roots.
Answer: False
D. Match the Following
| Column A | Column B |
|---|---|
| 1. D>0 | A. No real roots |
| 2. D=0 | B. Two distinct real roots |
| 3. D<0 | C. Two equal real roots |
| 4. b2−4ac | D. Discriminant |
| 5. ax2+bx+c=0 | E. Standard form |
Answers:
1–B, 2–C, 3–A, 4–D, 5–E
E. Assertion–Reason Questions
For each question, choose:
A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.
1.
Assertion: x2−5x+6=0 has two distinct real roots.
Reason: Its discriminant is positive.
Answer: A
2.
Assertion: A quadratic equation has no real roots when its discriminant is negative.
Reason: The square root of a negative number is not a real number.
Answer: A
3.
Assertion: x2+4x+4=0 has two equal roots.
Reason: Its discriminant is zero.
Answer: A
4.
Assertion: Every equation containing x2 is quadratic.
Reason: Simplification can sometimes cancel the x2 terms.
Answer: D
5.
Assertion: A negative root can sometimes be rejected in a word problem.
Reason: A mathematical root may not always satisfy the physical meaning assigned to the variable.
Answer: A
F. Very Short Answer Questions
1.
What is the standard form of a quadratic equation?
2.
What is meant by a root of a quadratic equation?
3.
How many roots can a quadratic equation have at most?
4.
What is the discriminant of ax2+bx+c=0?
5.
State the condition for two distinct real roots.
6.
State the condition for equal real roots.
7.
State the condition for no real roots.
8.
Write the quadratic formula.
9.
Why should an equation be simplified before deciding whether it is quadratic?
10.
What mathematical principle is used after factorising a quadratic into two factors?
G. Solve by Factorisation
1.
Solve: x2−7x+12=0
Answer: x=3,4
2.
Solve: 2×2+x−6=0
Answer: x=23,−2
3.
Solve: 3×2−10x+3=0
Answer: x=3,31
4.
Solve: 6×2−x−2=0
Answer: x=32,−21
5.
Solve: x2−10x+25=0
Answer: x=5,5
H. Discriminant-Based Questions
1.
Find the nature of the roots of: 2×2−3x+5=0
Answer: No real roots.
2.
Find the nature of the roots of: x2−6x+9=0
Answer: Two equal real roots.
3.
Find the nature of the roots of: 3×2−7x+2=0
Answer: Two distinct real roots.
4.
Find the discriminant of: 4×2+4x+1=0
Answer: 0
5.
For what value(s) of k will x2+kx+9=0
have equal roots?
Answer: k=±6
I. Conceptual / Thinking Questions
1.
An equation initially contains x3, but after simplifying, all x3 terms cancel and the resulting equation has degree 2. Can it be a quadratic equation? Explain.
2.
Why can a quadratic equation have at most two roots?
3.
A student says, “If I get a negative value of x, my calculation must be wrong.” Is this always true? Explain using the context of word problems.
4.
Without solving the equation, determine whether 5×2−2x+1=0
has real roots.
5.
Two quadratic equations have discriminants 25 and −4. What can you conclude about their roots without solving either equation?
Answers:
4. No real roots.
5. First has two distinct real roots; second has no real roots.
J. Word-Problem Questions
1. Rectangle
A rectangular plot has area 528m2. Its length is one metre more than twice its breadth. Form a quadratic equation for its breadth.
Answer:
If breadth =x, x(2x+1)=528 2×2+x−528=0
2. Consecutive Integers
The product of two consecutive positive integers is 306. Form the quadratic equation and find the integers.
Answer: 17,18
3. Ages
One person is 26 years older than another. Three years from now, the product of their ages will be 360. Form the quadratic equation and determine the younger person’s present age.
Answer: 9 years
4. Speed
A train covers 480 km at a certain speed. If its speed were 8 km/h lower, it would take 3 hours longer. Form the quadratic equation for the speed.
Answer: x2−8x−1280=0
Positive solution: x=40 km/h
5. Geometry
A right triangle has hypotenuse 13 cm. Its altitude is 7 cm shorter than its base. Find the two perpendicular sides.
Answer: 5 cm and 12 cm.
K. “Which Method?” Questions
1.
A quadratic is easily expressible as two linear factors. Which method is most convenient?
Answer: Factorisation.
2.
A quadratic is difficult to factorise directly. Which general method can always be used when real roots exist?
Answer: Quadratic formula.
3.
You only need to know whether real roots exist, not their actual values. Which quantity should you calculate?
Answer: Discriminant.
4.
A word problem gives a quadratic with two algebraic roots, one of which represents a negative length. What should you do?
Answer: Reject the root that is inconsistent with the physical situation.
L. Higher-Order MCQs
1.
For what value of k does x2−6x+k=0
have equal roots?
A. 3
B. 6
C. 9
D. 12
Answer: C
2.
Which equation has no real roots?
A. x2−4x+4=0
B. x2−5x+6=0
C. 2×2+2x+3=0
D. x2−1=0
Answer: C
3.
If the roots of a quadratic are 2 and 5, which equation has those roots?
A. x2+7x+10=0
B. x2−7x+10=0
C. x2−3x+10=0
D. x2+3x−10=0
Answer: B
4.
Which of the following has a repeated root?
A. x2−2x+1=0
B. x2−3x+2=0
C. x2+x−2=0
D. 2×2−5x+2=0
Answer: A
5.
If D>0, which statement is necessarily correct?
A. Both roots are equal.
B. There are no real roots.
C. There are two distinct real roots.
D. Both roots must be positive.
Answer: C
M. Exam-Style Mixed Questions
1.
Check whether the following equation is quadratic: (x+2)3=x3−4
Answer: Yes. On simplification, x2+2x+2=0
2.
Check whether x(x+1)+8=(x+2)(x−2)
is quadratic.
Answer: No. Simplification gives x+12=0
which is linear.
3.
Solve: 2×2−5x+3=0
by factorisation.
Answer: x=1,23
4.
Find the discriminant and nature of roots: 3×2−2x+31=0
Answer: D=0
Therefore, the equation has two equal real roots.
5.
A quadratic equation has discriminant 289. What does this tell you?
Answer: It has two distinct real roots.
⭐ Final Revision Challenge
Try these without looking at the answers:
- Find the nature of roots of 4×2−4x+1=0.
- Solve x2−9x+20=0.
- Find k if 2×2+kx+8=0 has equal roots.
- Form a quadratic equation whose roots are 4 and −3.
- A rectangle has area 180m2, and its length is 3 m greater than its breadth. Form and solve the resulting quadratic equation.
- Explain why x2+5=0 has no real roots.
- Give an example of an equation that looks cubic but becomes quadratic after simplification.
Answer key:
- Equal real roots
- 4,5
- k=±8
- x2−x−12=0
- Breadth =12 m, length =15 m
- Its discriminant is negative / x2=−5 has no real x
- Example: (x+2)3=x3−4 (after cancellation it becomes quadratic).