The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:
The real-world applications are highly dependent on the usage of mathematical equations. When you throw a stone, ball, or any other object in the air, it can travel slow or fast and traverse any distance, but how would you find its position at a particular time? Such things are estimated with the help of quadratic equations and their derivatives.
A quadratic equation is essentially an equation of second degree, where at least one of the terms is squared. The general structure of a quadratic equation (f (x)) is as follows:
ax2 + bx + c = 0; where ‘a’, ‘b’, and ‘c’ are constant values (known as coefficients of the equation) and ‘x’ is the unknown variable. For the equation to retain its quadratic nature, ‘a’ should never be zero.
Roots of a quadratic equation are the values of variables that satisfy the given quadratic equation. So, if ‘α’ is a root of the above-mentioned equation f(x), then f(α) = 0. The roots of the equation f(x) are (-b + √(b2 − 4ac))/2a, and (-b – √(b2 − 4ac))/2a.
Here, the expression (b2 − 4ac) is called the discriminant (D) of the equation. The discriminant decides the nature of the roots of the equation.
The roots of a quadratic equation can have the following characteristics depending upon the value of D:
Quadratic equations are used to measure many important entities like speed, distance, area, profit, etc. These equations simplify the estimations that need to be made in automotive, aerospace, and several technical industries on a large scale.
1. What are quadratic equations and functions?
Quadratic functions, also known as quadratic, are polynomials in which the highest-degree term carries the power of 2. A single variable quadratic function f (x) can be represented as:
f (x) = ax2 + bx + c = 0
The quadratic function is equated to zero to find the value of variables, which is then called a quadratic equation.
2. What are the three forms of a quadratic equation?
A quadratic equation can be expressed in three forms: standard, factored, and vertex.
The standard form refers the general expanded quadratic equation, as follows:
f(x) = ax2 + bx + c = 0
The factored form of quadratic equation is written in term of its roots, as given below:
f(x) = a(x – r1)(x – r2); where ‘r1’ and ‘r2’ are the roots of the equation.
The vertex form of a quadratic includes the horizontal and vertical coordinates ‘p’ and ‘q’, respectively. Since a quadratic graph is basically a parabolic curve, (p, q) is the vertex point of the parabola. The equation is illustrated as:
f(x) = a(x − p)2 + q
3. How do you solve quadratic equations?
There are various ways to solve a quadratic equation. It is commonly solved by factorization or completing the square.
Solution of a Quadratic Equation by Factorisation:
The basis of the factorisation method is that if r1 is the real roots of equation ax2 + bx + c = 0, then ar12 + br1 + c = 0. To find quadratics’ solution by factorization, you need to change its standard form into the factored form. After factorizing the equation, each factor is equated to zero to calculate the roots’ value. For example, suppose an equation is 6x2 – x – 2 = 0;
Here, a = 6, b = -1, and c = -2, you need to look for the factors of the product of ‘a’ and ‘c’, whose sum results in ‘b’. So, the factor form turns out to be (3x – 2)(2x + 1) = 0
Equating each factor to zero, i.e. on solving the linear equations, the value of roots is computed as 2/3 and -1/2.
This method works only if the equation has rational roots.
Solution of a Quadratic Equation by Completing the Square:
In this technique, the general form of equation is transformed into a square form using a few calculations. Hence, the equation ax2 + bx + c = 0 is modified to a(x + m)2 + n = 0; where m = b/2a and n = c – b2/4a. In simple words, you need to follow the given steps to solve the equation:
i. Divide all the terms of equation by ‘a’
ii. Take the term ‘c/a’ to the right side of the equation.
iii. On the left side of the equation, complete the square and then balance it by adding the same value to the right side.
iv. Perform the square root on both sides of the equation.
v. Subtract the constant that is left on the left side of the equation, and hence, you can calculate the value of x.
For example, if the equation is x2 + 4x – 5 = 0, you will get the following on completing the square: (x + 2)2 – 9 = 0. Thereafter, the roots (x) are found to be: -5 and 1.
4. What are the five examples of quadratic equations?
Some examples of quadratic equations in various forms and types are:
5. What is the easiest way to solve quadratic equations?
The roots of any quadratic equation (ax2 + bx + c = 0) can be found using the quadratic formula directly. Thus, x is calculated as:
x = (-b ± √(b2 − 4ac))/2a
To gain more in-depth conceptual knowledge of the quadratic equations along with solved examples, you can go through the video lectures on MSVgo.