JEE Main Study Notes for Nature of Roots of Quadratic Equations: The nature of roots of a quadratic equation ax²+bx+c=0 is determined by the discriminant, D =b²-4ac. If D=0, the equation has two real roots. If D<0, the equation has two complex roots. If D>0, the equation has two real and distinct roots.

• The topic Nature of Roots of Quadratic Equation from JEE Main Mathematics Syllabus includes subtopics of Identity, Completing the square, Discriminant, Repeated Roots, Factorization, and Formation of New Equations.
• In JEE Main, around 2-3 questions, for a total of about 8-12 marks are asked on the topic of Quadratic Equations.

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What is a Quadratic Equation?

• The name "quadratic equation" comes from the Latin word "quadratus," which literally means "square."
• A quadratic issue is one in which a variable is multiplied by itself, which is known as squaring in mathematics. The size of a square is equal to its side length multiplied by itself in this language.
• The second degree in x of an algebraic expression is called a quadratic equation. A quadratic equation has a constant form which is ax^2 + bx + c = 0, where a and b are coefficients, x is the variable, and c is the constant term.
• The coefficient of x^2 is a non-zero term(a 0), which is the first criterion for an equation to be a quadratic equation.
• The constant term is denoted by the letter ‘c,' whereas the linear coefficient is denoted by the letter ‘b,' and the quadratic coefficient is denoted by the letter ‘a.'
• Quadratic equations with only one unknown are referred to as univariate equations.
• Because they have non-integral powers of x, quadratic equations are often polynomial equations.
• They are known as second-degree polynomial equations since the maximum power is two.

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Roots of Equations

The roots of the equation are the values of the variable x that fulfill the equation in one variable. These roots could be simple or complicated. A quadratic polynomial becomes a quadratic equation when it is equal to zero. The roots of the quadratic equation are the values of x that satisfy the equation.

• A root of the quadratic equation ax^2+bx+c=0, a0 is a real number in general. If a^2+b+c=0, we can argue that x= is a quadratic equation solution. The roots of the quadratic equation ax^2+bx+c=0 and the zeroes of the quadratic polynomial ax^2+bx+c are the same.
• The quadratic equation ax^2+bx+c=0's roots are provided by
• x=–b±b^2–4ac√2a

Discriminant of Equations

The discriminant is a component of the quadratic formula, which is found below the square root. The discriminant of a quadratic equation is significant because it indicates the number and kind of solutions. This knowledge is useful since it acts as a double check when utilizing any of the four ways to solve quadratic equations (factoring, completing the square, using square roots, and using the quadratic formula).

The discriminant of a quadratic equation is calculated using the coefficients of the equation ax^2+bx+c =0 and is represented by the number D= b^2-4ac. The discriminant reveals the nature of an equation's roots.

The quadratic formula is used to find b^2 – 4ac.

The table below covers the many types of roots that are related to the determinant's values.

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For example, look at the quadratic equation y = 3x^2+9x+5. Find its discriminant.

Solution: The quadratic equation given is y = 5x^2-4x-3.

Discriminant formula is D = b^2-4ac.

This means, a= 5, b= 4 and c= 3

Therefore, the discriminant is,

D= 16-4.5.-3

=16+60

=76

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Nature of the Roots of Equations

The nature of the roots of the quadratic equation is determined by the discriminant's value, D=b^2–4ac.Depending upon D, then the quadratic equation's roots can be real or imaginary, depending on the criteria:

When b^2–4ac>=0, the roots are real, and when b^2–4ac < 0, the roots are fictional. Real roots are divided into two categories: rational roots and illogical roots.

• Rational Root

The roots of a quadratic equation are rational if ax^2+bx+c=0, where a,b,c are rational numbers, and b^2–4ac>0, i.e., D>0 and a perfect square.

• Irrational Roots

The roots are irrational if a quadratic equation is provided by ax^2+bx+c=0, where a,b,c are rational values and b^2–4ac>0, i.e., D>0 and not a perfect square. Using the discriminant approach, we may categorize the roots of quadratic equations into three kinds.

1. Two distinct real roots
2. Two equal real roots
3. No real roots

• Two Distinct Real Roots

There will be two roots in the quadratic equation ax^2+bx+c=0, which can be equal or unequal, real or unreal, or imaginary.

If D=b^2–4ac>0, we can get two separate real roots.

This can be depicted graphically as seen below. The graph of the quadratic equation intersects the x-axis at two unique locations, as seen in the graphical representation. The zeros or roots are these two separate points.

Example:

Take the quadratic equation x^2–7x+12=0

Here, a=1, b=7, and c=12 are the values.

D=b^2–4ac=(–7)²–4(1)(12)=1 discriminant

x^2–7x+12=0 has two unique real roots since the discriminant is bigger than zero.

The quadratic formula can be used to find the roots.

x=–(–7)±12×1=7±12

=7+12,7–12

=82,62

=4,3

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• Two Equal Real Roots

If D=b^2–4ac=0, we have two equal real roots in a quadratic equation ax^2+bx+c=0. The graph of the quadratic equation with equal roots only touches the x-axis at one point in the graphical depiction. The value of x is taken from this point. However, we can say that x has two equal solutions.

Example:

Consider the expression x^2–2x+1=0. D=b^2–4ac=(–2)2–4x1x1=0 is a discriminant.

x^2–2x+1=0 has two equal roots because the discriminant is 0.

To find the roots, the quadratic formula is used.

x= -(-2) +-0/2x1= 2/2=1

• No Real Roots

There will be no real roots in the quadratic equation ax^2+bx+c=0 if D=b^2–4ac0. Complex roots or imagined roots are the names given to the roots. 3x^2+x+4=0, for example, has two complicated roots: b^2–4ac=(1)2–434=–47, which is less than zero.

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JEE Main Previous Year Questions for Nature of Roots of Quadratic Equations

Given below are the previous year’s questions that came in JEE Main with solutions. Each question carries 4 marks.

Question 1: If equations x2+2x+3=0 and ax2+bc+c=0,a,b,c ∈ R, have a common root , then a:b:c is: (1) 1:2:3 (2) 3:2:1 (3) 1:3:2 (4) 3:2:1 [JEE-MAIN-2013]

Answer: (1)x2+2x+3=0D<0

Therefore, ax2+bc+c=0has both common roots

Hence, a:b:c=1:2:3

Question 2: Let ???? and ???? be the roots of equation x2-6x-2=0. If aₙ=????n- ????n , for n≥1, then value of (a₁₀-2a₈)/2a₉ is equal to : (1) 3 (2) -3 (3) 6 (4) -6 [JEE-MAIN-2015]

Answer: (1)Given x2-6x-2=0

Therefore, aₙ₊₂-6aₙ₊₁-2aₙ=0

(aₙ₊₂-2aₙ)/2aₙ₊₁=3

Now, put n=8

(a₁₀-2a₈)/2a₉= 3

Question 3: The sum of all real values of x satisfying the equation (x2-5x+5)x2+4x-60=1is : (1) 5 (2) 3 (3) -4 (4) 6 [JEE-MAIN-2016]

Answer: (2) x2-5x+5=1⇒x=1,4

x2-5x+5=-1⇒x=2,3but 3 is rejected x2+4x-60=0⇒x=-10,6

Sum = 3

Question 4: If ????, ????∈C are the distinct roots of the equation x2-x+1=0 , then ????¹⁰¹+ ????¹⁰⁷ is equal to - [JEE-MAIN-2018]

Answer: ????, ???? are roots of the x2-x+1=0

Therefore, ????= -⍵ and ???? = -⍵²

Where ⍵ is non-real cube root of unity

So, ????¹⁰¹+ ????¹⁰⁷

⇒(-⍵)¹⁰¹ +(-⍵²)¹⁰⁷

⇒ -[⍵²+⍵]

⇒ -[-1]=1

(As 1+⍵+⍵²=0 & ⍵³=1)

Question 5: If the cube roots of unity are 1, ω, ω2, then find the roots of the equation (x − 1)3 + 8 = 0. [JEE-MAIN- 2019 April]

Answer

(x − 1)3 = −8 ⇒ x − 1 = (−8)1/3

x − 1 = −2, −2ω, −2ω2

x = −1, 1 − 2ω, 1 − 2ω2

Question 6: Let α and β be the roots of equation px^2 + qx +r= 0, p ≠ 0. If p, q, r are in A.P. and 1⁄????+1⁄????= 4 then the value of |α -β| is: [JEE-MAIN- 2014]

Answer.

Let p, q, r are in AP ⇒ 2q = p + r ...(i)

Given 1⁄????+1⁄????= 4 ⇒

We have a + b = – q/p and ab =

4 ⇒ q = -4r ....(ii)

From (i), we have 2( – 4r) = p + r

⇒ p = –9r

q = – 4r

Now

Question 7: If x = a + b, y = aα + bβ and z = aβ + bα, where α and β are complex cube roots of unity, then what is the value of xyz? [JEE-MAIN-2017]

Answer

If x = a + b, y = aα + bβ and z = aβ + bα, then xyz = (a + b) (aω + bω2) (aω2 + bω),where α = ω and β = ω2 = (a + b) (a2 + abω2 + abω + b2)

= (a + b) (a2− ab + b2)

= a3 + b3

Tips for Preparation for Nature of Roots of Quadratic Equations

Mathematics has equal weightage to Physics and Chemistry in JEE, however, Mathematics has an advantage over the other two because if two or more candidates have the same overall score, the score in mathematics is used as a tie-breaker. JEE Main 2022 Math Syllabus contains a balance of difficult and easy topics. Given below are the tips to score well in Mathematics:

• Study all formulae- Knowing mathematical equations can be an added advantage on any math exam. It is important for the candidates to learn all the probability, geometry, trigonometry, and key calculus formulas. JEE Main study notes for the nature of roots of Quadratic Equations have been provided above for this purpose.
• Mathematical Applications- Candidates must revise differential equations and geometry applications. This holds true for properties of definite integrals as well as the outcomes of algebraic calculations of conic qualities.
• List of formulas- Making a list of formulas always helps candidates to keep it handy and revise. With everyday revision, it will get easier for candidates to remember all the mathematical formulas. A strong grip on formulas will help candidates solve math easily.
• NCERT Books- Candidates must refer to NCERT Books. Most of the topics are covered and questions are given from NCERT Books in JEE Main. While studying do maintain JEE Main study notes for the nature of roots of Quadratic Equations separately

Related Link: JEE Main Important Books

• Solve PYQs every day- Solving the previous year’s questions will help the candidates to get acquainted with the type of questions in JEE Main and better time management. Questions are sometimes repeated in JEE Main.
• Mock Test- Candidates must practice the Mock test two months before the examination. Math takes a lot of time to excel in. It is important for candidates to keep on giving mock tests.

Related Links :

• JEE Main Mock Tests
• JEE Main Test Series
• Revise- Candidates must follow a schedule that would help them to finish off the syllabus. After finishing off the syllabus, candidates must start revising. Revision helps in gaining confidence in the subjects. Revision helps in recollecting all the information studied.

Time Required for Preparation and Difficulty Level for Nature of Roots of Quadratic Equations

Nature of roots which is a sub-topic of quadratic equations can take up to 2-3 weeks to finish. Depending upon the calculation speed of the candidate, the topic can be moderate to slightly difficult. A table is provided below to provide an idea for the candidates.

JEE Main Nature of Roots of Quadratic Equations FAQs

Question: Which is the hardest topic in JEE Main 2022 math syllabus?

Answer: The hardest Topic in JEE Main 2022 Math syllabus are Complex Numbers, Probability and Statistics, Algebra, Trigonometry, Calculus, and Coordinate Geometry.

Question: How many questions are there in JEE Main 2022 Maths section?

Question: What will be the total weightage of Quadratic equations in JEE Main 2022 Exam?

Question: What is the negative marking in JEE Main 2022?

Question: What will be the mode of JEE Main 2022 examination?

Question: What is the general equation of the Quadratic Equation, a part of JEE Main 2022 Mathematics Syllabus?

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