Quadratic Equations Complete Tutorial - 2 - CAT 2009 Online Preparation

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CAT 2009 Online Preparation

Quadratic Equations Complete Tutorial - 2

Author: Sureshbala

Click Here for Quadratic Equations Complete Tutorial - 1

NATURE OF THE ROOTS

We mentioned already that the roots of a quadratic equation with real co-efficient can be real or imaginary. When the roots are real, they can be rational or irrational as also they can be equal or unequal. All this will depend on the expression b^2-4ac . Since b^2-4ac determines the nature of the roots of the quadratic equation; it is called the "DISCRIMINAN T" of the quadratic equation. A quadratic equation has real roots only if $b^2-4ac \geq 0$ .

if b^2-4ac<0 , then the roots of the quadratic equation will be imaginary.

Thus we can write down the following about the nature of the roots of a quadratic equation when a, b and c are all rational.

When	The roots are complex
When	The roots are rational and equal
When and a perfect square	The roots are rational and unequal
When But not a perfect square	The roots are irrational and (unequal)

Whenever the roots of the quadratic equation are irrational(a, b, c being rational) they will be of the form $a+\sqrt{b}$ and $a-\sqrt{b}$ whenever $a+\sqrt{b}$ is one root of a quadratic equation, then $a-\sqrt{b}$ will be the second root of the quadratic equation.

SIGNS OF THE ROOTS

We can comment on the signs of the roots, i.e. , whether the roots are positive or negative, based on the sign of the sum of the roots and the product of the roots of the quadratic equation. The following table will make clear the relationship between the signs of the sum and the product of the roots and the signs of the themselves.

Sign of product of the roots	Sign of sum of the roots	Sign of the roots
+ ve	+ ve	Both the roots are positive
+ ve	- ve	Both the roots are negative
- ve	+ ve	One root is positive and the other negative; the numerically larger root is positive
- ve	- ve	One root is positive and other negative; the numerically larger root is negative

CONSTRUCTING A QUADRATIC EQUAION

We can build a quadratic equation in the following three cases.

- when the roots of the quadratic equation are given

- when the sum of the roots and the product of the roots of the quadratic equation are given.

- when the relation between the roots of equation to be framed and the roots of another equation is given.

If the roots of the quadratic equation are given as $\alpha \, and \beta$ , the equation can be written as

$(x- \alpha)(x- \beta)=0 i.e x^2-x(\alpha+\beta)+\alpha \beta=0$

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-px+q=0

EQUATIONS OF HIGHER DEGREE

The highest power of x in the equation is called the degree of the equation. For example, if the highest power of x in the equation is 3, then the degree of the equation is said to be 3. An equation whose degree is 3 is called a cubic equation. A cubic equation will have three roots.

An equation whose degree is 'n' will have n roots.

CONSTRUCTING A NEW QUADRATIC EQUAION BY CHANGING THE ROOTS OF A GIVEN QUADRAIC EQUATION

If we are given a quadratic equation, we can build a new quadratic equation in the manner specified to us.

For example, let us a take a quadratic equation ax^2+bx+c-0 and let its roots be $\alpha \, and \beta$ respectively. Then we can build new quadratic equations as per the following patterns:

i) A quadratic equation whose roots are reciprocals of the roots of the equation ax^2+bx+c=0 the roots $\frac {1}{\alpha}$ and $\frac {1}{\beta}$

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

ii) A quadratic equation whose roots are k more than the roots of the equation ax^2+bx+c=0 , the roots are $(\alpha+K) \, and \, (\beta+k)$

This can be obtained by substituting (x - k) in place of x in given equation.

iii) A quadratic equation whose roots are k less than the roots of the equation ax^2+bx+c=0 , the roots are $(\alpha -k) \, and \, (\beta -k)$

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

iv) A quadratic equation whose roots are k times the roots of the equation ax^2+bx+c=0 , the roots are $k \alpha \, and \, k \beta.$

This can be obtained by substituting x/k in place of x in the given equation.

v) A quadratic equation whose roots are 1/k times the roots of equation ax^2+bx+c=0

the roots are $\frac {\alpha}{k}$ and $\frac {beta}{k}$

This can be obtained by substituting kx in place of x in the given equation.

MAXIMUM OR MINIMUM VALUE OF A QUADRATIC EXPRESSION

An equation of the type ax^2+bx+c=0 is called a quadratic equation. An expression of the type is called a "quadratic expression". The quadratic expression takes different values as x takes different values.