*Tuesday, August 18th, 2020*

As we all know a quadratic equation is a second-degree polynomial in x equated to zero and also if the co-efficient of x^{2 }is zero, it becomes a linear equation.

It would be a waste of time to explain how to find roots to a CAT aspirant so skipping the very normal basics letâ€™s understand the various types of question on quadratic that appears in CAT exam.

Very often we are required to obtain a quadratic equation when some conditions are given. The following conditions occur frequently.

- The roots of the quadratic equation are given.
- The sum of the roots and the products of the roots of the quadratic equation are given.
- The relation between the roots of the equation to be obtained and the roots of another equation is given.

**a.** If the roots of the quadratic equation are given as Î± and Î², the equation can be written as:

(x- Î±)(x- Î²) = 0 i.e., x^{2 }â€“ x(Î±+Î²) + Î±Î² = 0

**b.** If p is the sum of the roots of the quadratic equation and q is the product of the roots of the quadratic equation, then the equation can be written as x^{2 }â€“ px + q = 0

**c.** For this condition, we shall consider 5 possible relations.

If we are given a quadratic equation, we can obtain a new quadratic equation by changing the roots of this equation in the manner specified to us.

**i)** A quadratic equation whose roots are the **reciprocals **Â of the roots of the given equation ax^{2 }+ bx + c=0 i.e. the roots of the required equation are 1/Î± and 1/Î².

This can be obtain by substituting 1/x in place of x in the given equation giving us cx^{2 }+bx + a =0, i.e. we get the equation required by interchanging the co-efficient of x^{2 }and the constant term.

**ii)** A quadratic equation (say in y) whose roots are **k more** than the roots of the equation ax^{2 }+ bx + c=0, i.e. the rootsÂ are (Î± + k) and (Î² + k) or x+k=y

The required equation can be obtained by substituting (y-k) in place of x in the given equation.

**iii)** A quadratic equation (say in y) whose roots are **k less** than the roots of the equation ax^{2 }+ bx + c=0, i.e. the rootsÂ are (Î± – k) and (Î² – k) or x-k=y

The required equation can be obtained by substituting (y+k) in place of x in the given equation.

**iv)** A quadratic equation (say in y) whose roots are **k times** the roots of the equation ax^{2 }+ bx + c=0, i.e. the rootsÂ are kÎ± and kÎ² or kx =y

The required equation can be obtained by substituting y/k in place of x in the given equation

**v)** A quadratic equation (say in y) whose roots are 1/**k times** the roots of the equation ax^{2 }+ bx + c=0, i.e. the rootsÂ are Î±/k and Î²/k or x/k =y

The required equation can be obtained by substituting ky in place of x in the given equation.

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**Examples:**

**1.** Construct a quadratic equation from x^{2 }+ 7x + 16 = 0

**i)** 4 more than the current roots

**ii)** Reciprocal of the current roots

**Sol:**

** i.** For 4 more, as discussed in point ii. Replace x by (x-4) in the above equation. On solving we will get x^{2 }+ 15x + 16= 0

**ii.** For reciprocal of current roots, replace x by 1/x then on solving we will get the new equation as: 16x^{2}+ 7x + 1 = 0

**2.** Equation x^{2} + 5x â€“ 7 = 0 has roots a and b. Equation 2x^{2} + px + q = 0 has roots a + 1 and b + 1. Find p + q.

**Sol:Â **

There are two ways of solving it.

First way is to find out the roots then add 1 to them and then the form new equation. However, in this case the roots would be irrational so it is going to be an arduous task.

The another way is to form a new equation by replacing x to (x-1) {as roots are incremented by 1}.

On replacing x by (x-1), the new equation would be: (x-1)^{2} + 5(x-1) â€“ 7 = 0

On solving this you will get; x^{2} + 3x â€“ 11= 0; on multiplying by 2 we get: 2x^{2} +6x â€“ 22 = 0

On comparing , we get p =6 and q = -22.

Hence p+q= -16

Common Roots:

Condition for one common root:

Let Î± be the common root of the given equations a_{1}x^{2} +b_{1}x + c_{1}=0 and

a_{2}x^{2 }+ b_{2}x + c_{2}=0. Then the common root is given by:

Î± = (c1a2-c2a1)/(a1b2-a2b1) or Î± = (b1c2-b2c1)/(c1a2-c2a1)

**E.g:**

**a)Â **Find the condition that the equations x^{2} + px + q = 0 and

x^{2} + qx + p = 0 (pâ‰ q) have one root in common.

**Sol:** If one of the equations a_{1}x^{2} +b_{1}x + c_{1}=0 and a_{2}x^{2 }+ b_{2}x + c_{2}=0

have one root in common then (c_{1}a_{2 }– c_{2}a_{1})^{2 }= (b_{1}c_{2 }– b_{2}c_{1})(a_{1}b_{2}-a_{2}b_{1})

Here a_{1 }= a_{2 }=1

b_{1 }=p= c_{2 }

c_{1 }=q= b_{2 }

Therefore (q-p)2 = (p.p â€“q.q) (q-p)

â‡’ (q-p)2 = (p2 â€“ q2)(q-p)

â‡’Â q-p = (p-q)(p+q) â‡’ p+q = -1

**b)** If the equations x^{2} + 7ax + 10 = 0 and x^{2} + 3bx -85 = 0, where a and b are integers, have a common root, then the value of b would be?

**Sol:** Given the both equations have exactly one root in common, as the product of the roots of the two equations are different. This root must be rational.

The roots are rational, their sum in an integer and their product is rational therefore the roots are integers.

For equation (1) the roots are factors of 10 (Â±1, Â±2, Â±5, Â±10) ad for (2) they are factors of 85 (Â±1, Â±5, Â±17, Â±85) only if the roots are Â±2, Â±5

Suppose 5 is the common root; the other root of x^{2} + 3bx -85 = 0 is -17 then -3b = -17 +5

â‡’Â b =4

Suppose -5 is the common root; then the other root of x^{2} + 3bx -85 = 0 is -17 then -3b = 17 -5

â‡’Â b =4

Hence the value of b can be 4.

**c)** If a, b, c belong to R and equations ax2 + bx + c=0Â Â andÂ Â x2 + 2x + 9 =0Â have a common root, show that a:b:c = 1:2:9

**Sol:**Â Â The roots of the equation x2Â + 2x + 9 =0 are imaginaryÂ as the discriminant D= 4-36= -32 , which is negative.

As the imaginary roots of a quadratic equation are conjugate to each other, so both the roots of the equations will be common.

So, a/1 = b/2 =c/9

or,Â a:b:c = 1:2:9

The questions in CAT based on forming the equations by changing the roots of the current equation and on common are up to a limited extent especially the former one type. However, continue to practice the common roots questions by solving more questions.

Quadratic Equations based on x + 1/x type and other algebraic identities

Word Problems that can be solved using Quadratic Equation Techniques

Permutation and Combination â€“ Fundamental Principle of Counting

Permutation and Combination â€“ Distribution of Objects

How to find Rank of a Word in Dictionary (With or Without Repetition)

Set Theory- Maximum and Minimum Values

How to solve questions based on At least n in Set Theory for CAT Exam?

Sequence and Series Problems and Concepts for CAT Exam Preparation

Basic Probability Concepts for CAT Preparation

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