What Is Quadratic Function?
A quadratic function, also known as a second-degree polynomial function, is represented in the form of f(x) = ax2 + bx +c. Here, ‘a’, ‘b’, and ‘c’ are coefficients of real numbers, and the value pf ‘a’ is not equal to 0.
The most significant feature of the quadratic equation is that they are of the second order, or of degree 2. This implies that in all the quadratic functions, the greatest exponent of an independent variable “x” in a non-zero term is equal to 2. Visit Cuemath website to explore more about this topic in detail.
Quadratic Equation
A quadratic equation is a special case of a quadratic function. Here, the representation of a quadratic function is set equal to zero as shown below:
ax2 + bx + c = 0.
When all the values of constants ‘a’, ‘b’, and ‘c’ are known, we can use the quadratic equation to find the solution of ‘x’.
Forms of Quadratic Function
1. The standard form of quadratic function discussed above is represented as f (x) = ax2 + bx + c.
2. The factored form of a quadratic function is represented as f(x) = a(x-x1)(x-x2). Here x1 and x2 are the roots or zeroes of the quadratic equation. In these ‘x’ values the function crosses at the y-axis and the value of y is equal to zero.
3. The vertex form of a quadratic function is represented as f(x) = a(x-h)2 + k. Here both ‘h’ and ‘k’ are respectively the coordinates of the vertex. It is the point at which the function reaches either its maximum (when the value of ‘a’ is negative) or minimum (when the value of ‘a’ is positive).
Quadratic Formula
The quadratic formula states that for the given quadratic equation ax2 + bx + c = 0, the values of ‘x’ can be determined using the following quadratic formula:
In other words, the quadratic formula is used to determine the solutions or roots of the quadratic equation (ax2 + bx + c = 0) regardless of whether the roots are real numbers, complex numbers, whole numbers, and so on.
Criteria To Use Quadratic Formula
The two criteria to use quadratic formula are:
- The quadratic equation ax2 + bx + c = 0 must always be set equals to 0.
- The value of ‘a’ must not be equal to 0.
What are the Three Different Cases of Roots Derived using the Quadratic Formula?
As we know, the quadratic formula always delivers two roots, one when the (+ sign) is in the front of the square root and the other when (- sign) is in the front of the square root.
- The roots will be real and unequal if the discriminant (b2 – 4ac) is positive.
- The roots will be real and equal to each other if the discriminant (b2 – 4ac) is equal to 0.
- The root will be complex if the discriminant (b2 – 4ac) is negative.
Note: There will be two solutions when the discriminant is positive whereas there will be only one single solution when the discriminant is equal to zero.
Example: Solve the Quadratic equation 5x2 – 2x – 3 = 0 using the quadratic formula:
Solution:
Here,
a = 5
b = 2
c = 3
Using the quadratic formula:
As the discriminant is positive, the two values of x are -0.6 and 1.
