Mathematics Revision Notes of Quadratic Equation for NDA

Introduction

An equation of the form `ax^2 + bx + c = 0`, where `a != 0` and `a, b, c, x in R`, is called a real quadratic quation. The numbers a, b and c are called the coefficients of the equation. The quantity `D = b^2 - 4ac` is known as the discriminant of the equation `ax^2 + bx +c = 0` and its roots are given by `x = (-b pm sqrtD)/(2a)`.

An equation of the form `az^2 +bz +c = 0` where `a!= 0` and `a, b, c, z != C` (complex), is called a complex quad ratic equation and its roots are given by `z = (-b pm sqrtD)/(2a)`.

Nature of Roots of Quadratic Equation

Nature of roots of a quadratic equation `ax^2 + bx + c = 0` implies whether the roots are real or imaginary by analyzing the quantity D.

1. Let `a,b, c in R` and `a!=0` , then the equation `ax^2 + bx + c = 0` has

(i) real and distinct roots if and only if `D > 0.`
(ii) real and equal roots if and only if `D = 0.`
(iii) complex roots with non-zero imaginary parts if and only if `D < 0.` If `p + iq` (where `p, q in R, q != 0`) is one root of `ax^2 + bx + c = 0`, then second roots will be `(p - iq).`

2. If `a, b, c in Q` and D is a perfect square, then `ax^2 + bx + c = 0` has rational roots.

3. If `a, b, c in Q ` and `p + sqrtq` ( where, `p, q in Q`) is an irrational root of `ax^2 + bx + c = 0`, then other root will be `(p - sqrtq )`.

4. If `a = 1, b, c in I` are roots of `ax^2 + bx + c = 0`, are rational numbers, then these roots must he integers

5. If `ax^2+ bx + c = 0` is satisfied by more than two distinct complex numbers, then it becomes an identity i.e. `a = b = c = 0`.

6. If the roots of `ax^2 + bx + c = 0` are both positive, then the signs of `a` and `c` should be like and opposite to the sign of `b.`

7. If the roots of `ax^2 + bx + c = 0` are both negative, then signs of `a ,b` and `c` should be like.

8. If the roots of `ax^2 + bx + c = 0` are equal in magnitude but opposite in sign, then `b = 0` and `c < 0.`

9. If the roots of `ax^2 + bx + c = 0` are reciprocal to each other, then `c = a.`

10. In the equation `ax^2 + bx +c = 0` (where, `a,b,c in R`), if `a + b + c = 0` , then roots are `1, c/a` and if `a - b +c = 0` then the roots are `-1` and `-c/a`

Relation between Roots and Coefficients

`"1. Quadratic Roots"`

If `alpha` and `beta` are the roots of quadratic equation `ax^2 + b x +c = 0` , then

sum of roots `= alpha + beta = -b/a`

and product of roots ` = alpha beta = c/a`

`"2. Cubic Roots"`

If `alpha, beta ` and `gamma` are the roots of cubic equation `ax^3 + bx^2 +cx + d = 0 ; a != 0` ,then

`alpha + beta + gamma = -b/a`

`beta gamma + gamma alpha + alpha beta = c/a` and `alpha beta gamma = - d/a`

3. Biquadratic Roots

If `alpha, beta ,gamma` and `delta` are the roots of biquadratic equation `ax^4 + bx^3 + cx^2 + dx + e = 0 ; a != 0`, then

`alpha + beta + gamma + delta = - b/a`

`( alpha + beta ) ( gamma + delta ) + alpha beta + gamma delta = c/a`

`alpha beta ( gamma + delta ) + gamma delta ( alpha + beta ) = - d/a` and `alpha beta gamma delta = e/a`

`"4. Symmetric Roots "`

If `alpha ` and `beta` are the roots of quadratic equation `ax^2 +bx +c = 0` , then to find the symmetric function of `alpha` and `beta`, we use the following results.

(i) `alpha^2 +beta^2 = (alpha +beta)^2 - 2 alpha beta`
(ii) `(alpha -beta)^2 = (alpha + beta)^2 - 4 alpha beta`
(iii) `alpha^3 + beta^3 = (alpha + beta)^3 - 3 alpha beta ( alpha + beta)`
(iv) `alpha^3 - beta^3 = ( alpha - beta )^3 + 3 alpha beta ( alpha - beta )`

5. Common Roots (Conditions )

Suppose that the quadratic equations are `ax^2 +bx + c= 0` and `a' x^2 + b' x + c' = 0`.

(i) When one root is common, then the condition is `(a'c -ac')^2 = (bc' - b'c ) (ab' - a'b)`

(ii) When both roots are common, then the condition is `a/(a') = b/ (b')`

Formation of an Equation

`"1. Quadratic Equation"`

If `alpha` and `beta` are the roots of a quadratic equation, then the equation will be of the form

`x^2 - (alpha + beta ) x + alpha beta = 0`

`"2. Cubic Equation"`

If `alpha, beta` and `gamma` are the roots of the cubic equation, then the equation will be of the form

`x^3 - ( alpha + beta + gamma ) x^2 + ( alpha beta + beta gamma + gamma alpha) x - alpha beta gamma = 0`

`"3. Biquadratic Equation"`

If `alpha, beta, gamma` and `delta` are the roots of the biquadratic equation, then the equation will he of the form

`x^2 - (alpha + beta + gamma + delta ) x^3 + ( alpha beta + beta gamma + gamma alpha + alpha delta + beta delta + gamma delta ) x^2 - (alpha beta gamma + beta gamma delta + gamma delta alpha + delta alpha beta ) x + alpha beta gamma delta = 0`

Sign of Quadratic Equation

Let `f(x) = ax^2 + bx +c` or `y = ax^2 + bx +c`. where `a, b, c in R` and `a != 0`.

(i) If `a > 0` and `D < 0,` then f(x) > 0, `AA x in R`
(ii) If `a < 0` and `D < 0,` then .f(x) < 0 , `AA x in R`.
(iii) If `a > 0` and `D = 0`, then `f(x) >=0 , AA x in R`
(iv) If `a < 0` and `D = 0,` then. `f(x) <= 0, AA x in R`.

(v) If `a > 0, D > 0` and `f(x) = 0` have two real roots `alpha` and `beta` (where, `alpha < beta` ), then `f(x) > 0,` `AA x in (-oo, alpha) uu (beta ,oo)` and f(x) < 0, `AA x in (alpha , beta)`.

(vi) If `a < 0, D > 0` and .`f(x) = 0` have two real roots `alpha` and `beta` (where, `alpha < beta` ), then `f(x) < 0 , ``AA x in (-oo, alpha) uu (beta, oo)` and `f(x) > 0, AA x in (alpha , beta)`.

Quadratic expression and Its graph

In `y = ax^2 + bx + c`, if `a, b, c in R` and `a ne 0`. Graph of quadratic takes the shape of a parabola. The parabola opens upward or downward according as `a > 0` or `a < 0` respectively.

The lowest point `P` in figure-(i) and highest point Q in figure-(ii) is called as vertex of parabola. Now for different values of `a, b, c` if graph `y = ax^2 + bx +c` is plotted then following 6 different shapes are obtained.

Case-I

If `a > 0` and `D > 0`

Then quadratic equation has two roots and graph cuts the x-axis at two distinct points.

(i) For `a< x < beta => y` is negative.

(ii) For `x < a` or `x > beta => y` is positive.

Case-II

If `a > 0` and `D = 0`

Then curve touches x-axis. Hence both zeroes of polynomial coincides.

In this type equation becomes `y = a(x - a)^2` and `y ge 0`, for `x in R`.

Case-III

If `a > 0` and `D < 0`

Then curve completely lies above x-axis.

In this case imaginary roots appears and `y > 0` for `x in R`.

Case-IV

If `a < 0` and `D > 0`

Then graph is downward and cuts the x-axis at two distinct points.

In this case

(a) `y > 0`, if `a < x < beta`

(b) `y < 0`, if `x < a` or `x > beta`

Case-V

If `a < 0` and `D = 0`

Then graph touches the x-axis from below.

In this case `x in R, y le 0` for `x in R`.

Case-VI

If `a < 0` and `D < 0`

Then graph lies completely below the x-axis and `y < 0` for `x in R`.

Maximum and Minimum Value of `ax^2 + bx+c`

`ax^2 + bx + c = a(x+ b/(2a))^2 +((4ac -b^2)/(4a))`

Case I : If `a > 0`

Then, minimum value of `ax^2 + bx + c` is `(4ac -b^2)/(4a)` and this value occurs when `x = -b/(2a)` There is no maximum value when `a > 0`.

Case II : If `a < 0`

Then, maximum value of `ax^2 + bx + c` is ` (4ac-b^2)/(4a)` and this value occurs when `x = - b/(2a)` There is no minimum value when `a < 0`.

Method to Solve Fractional Quadratic Polynomial

Consider the fractional quadratic polynomial be `(a_1x^2 +b_1x+c_1)/(a_2x^2 +b_2x+c_2)`

Use following steps to solve it.

Step I : Equate the given expression to `y`

Step II : Obtain quadratic equation in `x` by simplifying the expression in step I.

Step III : Put discriminant `ge 0` of the equation which we get in step II.

Step IV : The values of `y` obtained by `D ge 0` is the solution set for the given rational expression.

Inequations

A statement involving one or more variables and sign of inequality `> , < , > ,` or `le` is called an inequation.

Note : For any real number a

`=> |x| le a leftrightarrow -a le x le a`

`=> |x| ge a leftrightarrow -x le -a ` or `x ge a`

Solution of Quadratic Inequations

Let `f(x) =ax ^2 + bx + c`, where `a, b, c in R` and `a ne 0`.

Then, `f(x) ge 0, f(x) > 0, f(x) le 0, f(x) < 0` are called quadratic inequations.

The set of real values of `x`, which satisfy the inequation, is called the solution set.

Solution of Linear Inequations in Two Variables

In order to represent the solution set of linear inequation in two variables, we follow the following steps,

`"Step I :"` Convert the given in equation say `ax + by le c` into the equation `ax +by =c` and the graph.

`"Step II :"` Choose a point not lying on this line `ax +by= c` substitute its coordinates in the inequation. H the inequation is satisfied, then shade the portion of the plane which contains the chosen point, otherwise shade the portion which does not contain the chosen point.

`"Step III "`: The shaded region obtained in step II represent the desired solution set.

 
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