What is Quadratic Equation, Different Forms & How To Solve

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The word “quadratic,” which describes how the variable is squared in the equation, is derived from the Latin word “quadratus,” which means square. Put differently, a “equation of degree 2” is a quadratic equation. Quadratic equations are used in various fields of study, including maximising or reducing the area of an enclosed space given a perimeter or even the trajectory of a ball during a sporting event. In this write-up, we will discuss what is quadratic equation, different forms of it and how to solve it. 

What is Quadratic Equation?

A quadratic equation is a polynomial equation that contains the square of an unknown variable, commonly represented by x. Quadratic equation is written in ax2 + bx + c = 0 format. In this equation, x is the variable and the rest are constant. The a, b and c values may be numbers or coefficients. An example quadratic equation would be: 

x2 + 5x + 6 = 0

The highest degree that the variable x is raised to in a quadratic equation is two. If you are searching for what is quadratic equation, 

you must know that the reason why these equations are called quadratic is due to the fact they include a squared time period. Quadratic equations commonly have two answers or roots, that are the values you can plug in for x that make the equation genuine.

Different Forms of Quadratic Equation 

Now that you know what is quadratic equation

let’s have a look at different forms of it– 

1.The Standard Form

The most commonly identified form that a quadratic equation is written in is the standard form that is  ax2+bx+c = 0

Where a, b and c are actual numbers and x is an unknown variable. This shape displays the total quadratic expression, with the x2 term, the x time period, and the constant like x2 +5x+6 = 0  

2.The Factored Form

When we write quadratic equation in factor form, it looks like this:

a(x – r1)(x – r2) = 0

Where r1 and r2 are the solutions or roots of the equation. This form reveals the two linear factors that make up the quadratic expression. For example:  

(x – 3)(x – 2) = 0

The roots can be figured out by setting each individual factor equal to 0 and solving. The factored form displays the solutions of a quadratic in a more direct way.

3.The Vertex Form 

The vertex form of a quadratic equation generally written as- 

a(x – h)2 + k

Where (h, k) represents the vertex point of the graphed quadratic function. For example:

a(x – 3)2 + 2  

This form reveals the minimum or maximum point (vertex) of the parabolic graph visually as the point (h, k). Using vertex form lets you quickly identify the vertex and opens/direction.

How To Solve Quadratic Equation?

There are three main ways to solve quadratic equations to find the roots or solutions – factoring, using the quadratic formula, and completing the square.

1.Factoring

If you want to know what is quadratic equation and how to solve it, factoring is one of the most commonly used methods. If a quadratic equation can be factored, it is also the fastest solving method. You break down the equation into linear factors:

(x – a)(x – b) = 0

After that you need to equal each factor to zero to solve and you will get  x = and x = b. 

For example, to solve x2+ 5x + 6 = 0:

Factor: (x + 2)(x + 3) = 0  

Solutions: x = -2, x = -3

2.Quadratic Formula 

If we can’t apply factoring, we use the quadratic formula that is:  

x = (-b ± √(b2 – 4ac)) / 2a

To solve:

1) Identify a, b, and c  

2) Plug into formula

3) Solve

For example, to solve 6×2 – 17x +12 = 0:

x = (-17± √(-17)2 – 4(6) (12)/ 2(6)

By solving it, we get x=3/2 and 4/3.

3.Completing the Square

This method creates a perfect square trinomial, then solves by taking square roots. It is used to determine the vertex form. For example, to solve x2 + 6x + 7 = 0, take half of 6 which is 3, square it to 9, then add and subtract 9 inside the parentheses:

x2 + 6x + 7 = 0

(x + 3)2 – 9 + 7 = 0

(x + 3)2 – 2 = 0

√(x + 3)2 = √4

x + 3 = ±2

x = -1

So the solution is x=-1, found by completing the square into a trinomial and simplifying.

Conclusion

Knowing what is quadratic equation and solutions is essential foundation in algebra and higher maths concepts. Quadratic equation applications include finding its roots, analysing the discriminant to determine nature of solutions, and linking equation components to graphical behaviour. When you know the concept of quadratic equation, you will be able to comprehend more advanced polynomial, exponential, logarithmic and rational concepts that need quadratic knowledge as a base. 

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