Quadratic Equations
A polynomial equation of second degree is called a Quadratic Equations.The standard form of the equation is ax ^{2} + bx + c = 0 where a,b and c are constants and a≠0.
The exponent of the x should be an integer.
Examples:
1) x ^{2} + 5x + 10 = 0
Solution : It is a quadratic equation as the degree of the equation is 2.
2) 4a ^{2} + 5a - 15= 0
Solution : It is a quadratic equation as the degree of the equation is 2.
3) 2/3p ^{2} - 2p + 20 = 0
Solution : It is a quadratic equation as the degree of the equation is 2.
Example :
x ^{2} + 2x ^{1/2} + 4 = 0 is not a quadratic equation as it contains the term x ^{1/2} , where ½ is not an integer.
The constant ‘a' assuming the value zero reduces the equation to linear form.
When we graph such equation, we get a parabola.
Example : f(x) = x ^{2} + 6x + 8
The above graph is a upward graph with lowest point known as the Vertex of the parabola whose coordinates are ( -3, -1).
The points where graph cut the X- axis represent the zeros of the function. The X- coordinates give the roots of the equation.
Note : Roots of a Quadratic Equation : Let P(x) = 0 , be a quadratic equation, then the zeros of the polynomial P(x) are called the roots of the equation P(x) = 0
Thus x = α is a roots of P(x) = 0 if and only if P(α)= 0
Different forms of the quadratic equations :
1) | Standard form : a^{2}+ bx + c =0 |
Used to find roots of the equation. |
2) | Vertex form a(x-h)^{2} + k |
Used to draw a parabola |
3) | Factored form : a(x - x1)(x - x2)= 0 |
Roots are x1 and x2 |
There are different methods to factoring the Quadratic Equations.
Check whether the following equations are quadratic or not?
1) 4x ^{2} - 4 = 0
Solution :
It is a quadratic equation since the highest degree is 2.
2) x - 1/x = x ^{2}
Solution :
x - 1/x = x ^{2}
(x ^{2} - 1)/x = x ^{2}
∴ x ^{2} - 1 = x ^{3} [ cross multiply]
∴ -x ^{3} + x ^{2} - 1 = 0
The degree of the equation is 3 so its not a quadratic equation.
3) 5 - x(x - 3) = 0
Solution :
5 - x(x - 3) = 0
5 - x ^{2} + 3x = 0
The degree of the equation is 2 so its a quadratic equation.
According to the equation, a=-1,b=3 and c=5
the axis of symmetry is x = x-coordinate of vertex.
x-coordinate of vertex = -b/2a
x = 3/(-2 x 1)
x = -3/2
x = -1.5
∴ Equation of axis of symmetry is, x = -1.5
Introduction of Quadratic Equations
• Splitting of middle term
• By completing the square
• Factorization using Quadratic Formula
• Vertex form from Quadratic Equation
• Finding Axis of Symmetry in Quadratic equation
• Solved Problems on Quadratic Equation