Formation of Quadratic Equations with Given Roots

Introduction 

Aspirants aiming to crack the JEE Mains exam even start their preparation from their class 11th so that they would have time to grasp and have a thorough understanding of every concept. Considering the fact that IIT JEE Mains is one of the toughest exams in India, which requires unmatched hard work and perseverance, going through every chapter or topic 8-10 times is no big deal. 

Talking about the JEE Mains topics, one of the most important topics in maths is the formation of quadratic equations with given roots, which plays a significant part in the IIT JEE Mains exam as a lot of questions are being asked every year from this.  When it comes to national-level examinations, every mark matters; therefore, there’s no way you can leave any topic. To help you in your preparation, Unacademy has come up with easy-to-understand solutions which will boost your knowledge every time. So, make sure you stick by us till the end! Without any further delay , let’s get into it! 

What is a Quadratic Equation?

A quadratic equation is also known as the quadratics, and is referred to as the second-degree polynomial equation, which states that there is at least one term, which is squared, the quadratic equation is written as the  f(x) = ax2+ bx + c. Here we can say, a ≠ 0 and a, b, c, ∈ R. According to this equation, ‘a’ is termed as the leading coefficient whereas the constant  term of f (x) is c. The x values which satisfy the quadratic equation are also the quadratic equation roots given as (α,β). Note, there will always be two roots of a quadratic equation, however, their nature can be either imaginary or real. 

If the quadratic polynomial is equal to  zero, it forms a quadratic equation. The x value which satisfies the equation is referred to as the roots of the quadratic equation. 

The quadratic equation general form of roots can be written as ax2+ bx + c = 0

Example of the roots of the quadratic equation is  3×2 + x + 5 = 0,  -x2 + 7x + 5 = 0, x2 + x = 0.

The formula of Quadratic Equation 

The quadratic equation formula is described as  – 

(α,β) = – b ±b2-4ac2a

Important Formulas for Solving Quadratic Equation 

Below, we have listed some of the most important formulas used to solve a quadratic equation. Let’s have a look- 

1.The roots of the quadratic equation: x = – b ±b2-4ac2a, where D = b2 – 4ac

2. Nature of roots:

If D > 0 then roots are real and distinct (unequal)

If D = 0 then roots are real and equal (coincident)

If D < 0 then roots are imaginary and unequal

3. The roots ‘α+iβ’ and ‘α-iβ’ are the conjugate pair of each other.

4. Sum and Product of roots: If α and β are the roots of a quadratic equation, then

Sum of roots  = S = α+β = -b/a = – coefficient of xcoefficient of x2

Product of roots = P= αβ = c/a =  constant termcoefficient of x2

5. Quadratic equation in the form of roots:   x2 -( α+β )x + αβ =0

6. The quadratic equations a1x2+b1x+c1 = 0 and a2x2+b2x+c2 = 0 have 

One common root if  (b1c2 -b2c1) (a1b2 -a2b1)= (a1c2 -a2c1)2

Both roots common if a1a2 = b1b2 = c1c2

7. In quadratic equation  ax2+ bx + c = 0

If a > 0 , minimum value  is 4ac -b24a at x = -b2a

If a < 0 , maximum value  is  4ac -b24a at .x = -b2a

8. If α ,β , are roots of cubic equation ax3+ bx2 +cx + d = 0 , then, α +β + = -ba ; αβ+β +α = ca  and αβ =  -da

9. A quadratic equation becomes an identity (a,b,c =0) if the equation is satisfied by more than two numbers i.e. having more than two roots or solutions either real or complex.

Roots of Quadratic Equation 

The variable values which satisfy the given quadratic equation are referred to as their roots. In simple terms, we can say that the quadratic equation f(x) has a root  x = α, only if f(α) = 0. 

• If one of the quadratic equation roots is  zero then  c = 0
• If b = c = 0, both the quadratic equation roots will be zero. 
• if a = c, then the quadratic equation roots are the reciprocal of each other

Nature of Roots of Quadratic Equation 

Let’s study the nature of the root of quadratic equation in detail –

Formation of Quadratic Equations with Given Roots

Let’s understand the formation of quadratic equations with given roots. 

Imagine there are two different roots of a quadratic equation given as α and β then  x2 – (α + β)x + αβ  =  0 will be the formula to build the quadratic equation. 

This also means, 

 x2  – (sum of roots)x + product of roots = 0

Imagine the quadratic equation’s standard form is ax2 + bx + c  =  0 where a, b, and c are  real numbers. Imagine, α and β  are the zeros of the quadratic equation mentioned above. 

Here is the formula to get the quadratic equation roots product and sum- 

α+β = -b/a = – coefficient of xcoefficient of x2

αβ = c/a =  constant termcoefficient of x2

Conclusion 

Now, when you have understood everything about the quadratic equations, it’s time to jump onto another topic.

Our today’s article is about the formation of quadratic equations with given roots, what exactly quadratic equation is, roots of quadratic equation, quadratic equation formula, and other topics related to the same ends here. Since class 10th, we have been studying the significance of quadratic equations in mathematics, and even for national level examinations like IIT, this topic plays an important role as several questions are being asked from this topic every year.