An expression which consists of constant, variables and exponent values is known as polynomial expression. The polynomial expression is combined with the addition, subtraction, and multiplication.
Analyzing Graphs of Quadratic EquationsBack to Top
Domain of the above function f (X) is the Set of all Real Numbers. y- intercept of graph of f (X) is given by f (0) = c and x- intercepts are obtained by solving the equation JX2 + KX + L = 0 using the Quadratic Formula.
Let’s understand the process of analyzing graphs of quadratic Functions. Range of quadratic function f (X) can be found as shown below:
First, f (X) = J (X - h) 2 + C,
f (X) = JX2 – 2JhX + Jh2 + C
Two conditions must be followed -2Jh = K (first equation) and Jh2 + C = L (second equation). From first equation, h = -K / 2J. Using second equation,
C = L - K2 / 4J
The term (X - h) 2 is either positive or zero, hence (X - h) 2 ≥ 0.
If J > 0, multiply both sides of inequality above by ‘J’
J(X - h) 2 ≥0,
Add ‘C’ to both sides of inequality
J(X - h) 2 + C ≥ C,
Thus f (X) ≥ C
Thus range of f (X) is given by
(C, +∞). Since minimum value of f (X) is C and it can be concluded that range of f (X) can also be given by (-∞, C).
From above discussion, it is clear that graph of quadratic function is called as Parabola and Point with coordinates (h, C) is called vertex of parabola.
Consider the equation f (x) = 2x2 + 2 x - 4, for which graph looks like this:
Solving Quadratic Equations by FactoringBack to Top
Let’s discuss process of solving Quadratic Equation by factoring. Let’s consider the following quadratic equation to understand how to solve quadratic equations by factoring?
A y2 + B y + C = 0,
Since maximum degree of variable 'y' is two (2), hence this type of equation is referred to quadratic equation. A, B and C are called quadratic coefficients. There are two methods to solve a quadratic equation. One is direct Factorization and second is using Quadratic Formula.
Above equation can be solved using quadratic formula which is given as:
This is the standard formula for factorizing quadratic equation.
Let’s take following example:
Suppose we are given P2 + 9P + 20 =0 then what is the solution of this equation? There are two types of factorization methods. One method includes direct factorization. In direct factorization, we multiply first (P2) and last (20) terms which gives a new value (20P2). Then distribute the middle value (9P) in such a way that its multiplication gives (20P2) and addition or subtraction gives middle value. Another method is use of quadratic formula.
Using the first method, we get following solution:
P2 + 9P + 20 = P2 + 5P + 4P + 20 = 0,
P (P + 5) + 4 (P + 5) =0,
(P + 5) (P + 4) = 0,
Then solution will be,
P = -5 or P = -4,
Using quadratic formula same results can be obtained. This is all about factoring quadratic equation.
Quadratic InequalitiesBack to Top
1.) ax2 + b x +c > 0, example: x2 + 4x > 5,
2.) ax2 + b x +c < 0, example: 8x2 < 29,
3.) ax2 + b x +c < 0, example : 6 ≥ x2 – x,
4.) ax2 + b x +c <= 0, example : 4y2 + 1 ≤ 8y,
There are some steps for solving quadratic inequalities :->
Step 1) First of all we have to move all terms to one side.
Step 2) Factorize the inequality.
Step 3) After this find the roots of this quadratic equation.
Step 4) We can graph quadratic inequalities by plotting coordinates on graph and then shading appropriate area according to inequality sign.
Let’s take an example: x2 + 4x < 5,
Solution: First of all we will move 5 to left side
x2 + 4x – 5 < 0, now solve this equation and find the root, so, x2 + 5x - x - 5 <0 => x (x + 5) - 2 (x + 5) < 0 ,
=> (x - 2) (x + 5) < 0, so x = 2, - 5.
Let’s take the value x = 0 then 0 – 0 -5 <0
Now x = 1 , 1 + 4 -5 < 0
Now x = -4 => 16 – 16 -5 <0
Now x = -3 => 9 – 12 – 5 < 0
So it is proved that x2 + 4x – 5 < 0 in interval ( 2 -5).
Quadratic Formulas and DiscriminantBack to Top
Discriminant = b2 – 4ac.
Suppose a quadratic equation x2 + 4x + 4 = 0, here a = 1 ,b = 4 ,c = 4 so discriminant is 42 – 4ac => 16 – 4 . 1 . 4 => 16 - 16 = 0.
We can also find the nature of solution using this discriminant .
Case 1) If value of b2 – 4 a c >0 means discriminant is positive discriminant, in this case two real solutions are generated, if discriminant is a perfect Square then roots are rational, otherwise they are irrational.
Case 2) If value of discriminant is zero, b2 – 4 a c = 0, then there is only one real solution.
Case 3) If value of discriminant is negative, b2 – 4 a c < 0, then in this case there is no real solution, there will be two imaginary solutions.
Example:- Calculate the discriminant, nature and number of solutions of equation y = x2 + 4x + 5, in this equation a = 1, b = 4, c = 5, so discriminant = b2 – 4 a c?
Put value of a, b, c in this formula, then (42 – 4 .1.5) => 16 – 20 = -4.
Here discriminant is negative, so there are no real solution for this quadratic equation, solutions are imaginary. So using Quadratic Formula for y = x2 + 4x + 5,
X1 = (- (4) +- √(42 – 4 .1.5))/2.1
= (4 - √(-4)) /2,
= (4 + 2 i)/2, (4 - 2 i)//2,
= 2 + i, 2 - i.
Solving Quadratic Equations by GraphingBack to Top
⇨ x = - b + √ (b2 – 4ac) / 2a, its alternate form is given by:
⇨ x = 2c / -b +√ (b2 – 4ac).
Let’s understand how to solve quadratic equation by Graphing. To solve quadratic equations by graphing method we need to follow some steps:
Step 1: Let quadratic equation is p2 – 8p + 15 = 0.
Step 2: Now with help of above defined formula we can find the factors. As we know that quadratic formula is given as: ⇨ x = - b + √ (b2 – 4ac) / 2a. In equation value of ‘a’ is 1, value of ‘b’ is -8 and value of ‘c’ is 15.
Step 3: Put these values in formula. On putting these values we get:
⇨ x = - b + √ (b2 – 4ac) / 2a,
⇨ x = - (-8) + √ ((-8)2 – 4 * 1 * 15) / 2 (1),
⇨ x = 8 + √ (64 – 60) / 2,
⇨ x = 8 + √ 4 / 2, so here we will get two factors of given equation.
X = 4 + √ 1 and X = 4 - √ 1. Now using these factors we can plot quadratic equation graph. The quadratic equation are shown below:
This is all about quadratic equation graph. This is the process of Solving Quadratic Equations by graphing.
Solving Quadratic Equations by Completing SquaresBack to Top
In mathematics completing the Square means to find the last term of perfect square trinomial. When we square the binomial term we get a trinomial expression. Meaning of squaring the binomial term is that we multiply the term to it. Let us see some examples of perfect trinomial square:
If we have a binomial as x + 1, when we find the square of binomial term we get a trinomial expression which is (x + 1)2 = x2 + 2x + 1.
In order to complete the square we will find the square of binomial. There is a particular formula to get the square of binomial.
(a + b)2 = a2 + 2ab + b2,
General form of quadratic equation is ax2 + bx + c.
When we compare these two equations we find that 'a' is a variable and 'b' is a Integer value. 'C' is half of square of 'b'.
(x + 3)2 = x2 + 2(3)x + 32,
(x + 3)2 = x2 + 6x + 9
Here 'a' is a variable and 'b' is an integer. Value of c = ½ (b)2 which is ½(3)2 = 9.
General formula of a perfect square trinomial is:
(x + p/ 2)2 = x2 +2 (p / 2)x + (p / 2)2.
This is the method by which we can complete a square.