Quadratic equations

Introduction

Have you seen a pendulum? Or a rocket being launched? You might have played on the swings. Or perhaps thrown a ball in the air. Do you know what they have in common? In all these examples, the path of the objects can be described by a quadratic equation.

The word quadratic is derived from the Latin word quadratus, which means square. So quadratic equations are algebraic expressions where the highest exponent or the power is 2. This is why quadratic equations are also called the equations of degree 2. This topic is one of the most crucial parts of mathematics, and most of the topics in mathematics are directly or indirectly associated with quadratic equations. It is also very important from the IIT JEE examination point of view; last year, 5-6 direction questions and 2-3 application-based questions were reported from this chapter. Do not skip this chapter if you want to qualify for your exam with flying marks.

Definition of Quadratic Equation

Quadratic equations are polynomial equations whose highest exponent or power is two. A standard quadratic equation looks like this:

ax² + bx + c = 0 

where x is variable, a and b are coefficients, c is the constant, and a ≠ 0. 

The quadratic is univariate, which means it has only one variable. The equation is polynomial because the variable x has non-negative integer powers.

In a quadratic equation, the variable x has two values, obtained after solving the equation. These two values are also called the roots or zeros of the equation. 

The formula of Quadratic Equation

Because quadratic equations have a degree of two, the equation will have two solutions. If the quadratic equation is ax² + bx + c = 0, then the formula to get the roots of this equation is:

x = -b ± b² – 4ac^2a

There will be two solutions for x, as shown by the plus/minus symbol. The quadratic equation formula is explained here.

Examples of Quadratics

The quadratic equations of the form (ax^2 + bx + c = 0) are shown below.

1. x^2-3x+4=0 here, a=1, b=3,c=4
2. 3x^2+9=0 here, a=3, b=0,c=9
3. 4x^2+x-7=0 here, a=4,b=1,c=7
4. x^2+4x=2 here a=1, b=4,c=2
5. 2x^2-3=0 here a=2, b=0,c=3

Roots of Quadratic Equation

The roots of a quadratic equation are the values of variables that satisfy it. 

The x-coordinates of the sites where the curve y = f(x) intersects the x-axis are the real roots of an equation f(x) = 0.

• If c = 0, one of the quadratic equation’s roots is zero, while the other is -b/a
• If b = c = 0, both roots are zero
• If a = c, the roots are reciprocal to one other

How to Solve a Quadratic Equation?

Quadratic equations may be solved using one of four methods:

• Graphing: This is an effective visual strategy. You can figure out where the quadratic function f(x) = ax² + bx + c crosses the x-axis by graphing it. The big drawback of this method is that finding the correct values of x might be challenging. In addition, if the answers are complicated, the graph will not cross the x-axis (in the case of a negative discriminant)
• Factoring: In certain circumstances, this technique may save time by avoiding the need to graph, complete the square, or use the quadratic formula. However, certain quadratics are tricky to factor. In such cases, the quadratic formula might be more helpful
• Complete the square: This procedure is lengthy but effective. This procedure may be simplified using the quadratic formula. The method of completing the square is also used to convert some circular equations to their appropriate form
• Quadratic formula: This strategy is always effective. It’s calculated by applying the preceding approach (complete the square) to a generic quadratic in standard form, ax² + bx + c = 0. The main disadvantage is that the computations, which include multiplication, addition, radicals, and division, may get laborious (fractions)

The range of quadratic equations

The range of quadratic equation of a function y = f(x) is the set of values y takes for all values of x within the domain of f. The graph of any quadratic function, of the form f(x) = ax²+bx+c, which may be expressed in vertex form as follows:

f(x) = a(x – h)² + k , where h = – b / 2a and k = f(h)

When a > 0, it is either a parabola opening up or a < 0, it is a parabola opening down.

• As a result, if a > 0, f’s graph has a minimum point, and
• If a < 0, the graph of f has a maximum point. The vertices of parabolas with coordinates (k, k) where h = − b / 2a and k = f are both minima and maxima (h)

Conclusion

Above, we have studied the quadratic equation-definition, formula, roots, and examples. We have gone through the standard formula of the quadratic equation, which is ax² + bx + c = 0, where x is variable, a and b are coefficients, c is the constant, and a ≠ 0. This chapter has a unique value in our mathematics and physics subjects as most topics like calculus, Linear programming, vectors, etc., have many questions based on this topic.So, it is advised to not skip this topic for your upcoming examinations.