Quadratic Equations

A quadratic equation is a second-degree polynomial equation written as:
ax² + bx + c = 0

Where:
a, b are coefficients (with a ≠ 0),
x is the variable
c is the constant
And the highest power of x is 2.

The word ‘quadratic’ is derived from the Latin word ‘quadratus’, which means square. The equation is quadratic because ‘quad’ means the square (power 2). The solutions to a quadratic equation, known as its roots, can be real or complex numbers. The graph of a quadratic equation is known as a parabola.
 

The roots of a quadratic equation of the form ax2 + bx + c = 0 are values of x that equate the equation to zero. Hence, they are also called zeros, and they satisfy the equation. Solving a quadratic equation yields two values of x, which can be real or complex. For e.g., the equation x2 - 3x - 4 = 0 has two roots; x = -1 and x = 4. 

We can verify this by substituting the values of x in the equation:

When x = -1, we get: (-1)2 - 3(-1) - 4 = 0
Simplifying it: 1 + 3 - 4 = 0 
4 - 4 = 0

When x = 4, we get: (4)2 - 3(4) - 4 = 0 
Simplifying it: 16 - 12 - 4 = 0 
4 - 4 = 0

There are various methods to find the roots of a quadratic equation. One of them is by using the quadratic formula.

There are a few quadratic equations that cannot be factorized with ease, and here use this formula to find the roots. The two roots in the quadratic formula are written in the form of a single expression. Solving a quadratic expression will usually yield two roots; one with a positive value and one with a negative value.

The quadratic formula for the equation ax2 + bx + c = 0 is: 
x = -bb2-4ac2a.

Example: Let’s find the roots of the quadratic equation y² + 2y - 15 = 0 using the quadratic formula.

Here,
a = 1,
b = 2,
c = −15
Using the formula:  y = -b b2-4ac2a

Substituting the values: y = -2  (2)2- 4(1) (-15)2(1)
                              
                                         = -2   4 + 602

                                         = -2   642
                                         

                                         = -2  82

So,                 
                 y = -2 + 82 = 62 = 3
        
       and,   y = -2 -82 = - 102 = -5

Therefore, the roots of the equation are 3 and -5.
 

The quadratic formula, used in solving equations of the form ax2 + bx + c = 0, is obtained by following the process of completing the square. In this manner, we convert the given equation into the perfect square form, and thus we can isolate the variable and get a general solution that is applicable for any given coefficient.

Example: Examine the following arbitrary quadratic equation: ax2 + bx + c = 0, a ≠ 0.

We take the following steps to find the equation's roots:

Step 1: ax2 + bx = -c ⇒ x2 + bxa = -ca

By adding a new term (b/2a)2 to both sides, we can now represent the left-hand side as a perfect square:

x2 + bxa + (b2a)2 = -ca + (b2a)2.

Now, the left side is a perfect square:

(x + b2a)2 = - ca + b24a2 ⇒ (x + b2a)2 = (b2 - 4ac 4a2)

Simplifying the equation further, we get:

x + b2a = ± b2 - 4ac 2a

x = (-b ± √(b2 - 4ac)2a

We were therefore able to isolate x and derive the equation's two roots by completing the squares.
 

Greek letters alpha (α) and beta (β) are commonly used to represent the roots of a quadratic equation. Here, we'll learn more about determining the type of roots in a quadratic equation.

It is possible to determine the nature of a quadratic equation's roots without actually determining the equation's roots (α, β). This can be accomplished by calculating the discriminant value, which is a component of the quadratic equation solution formula. The discriminant of a quadratic equation, denoted by the letter ‘D,’ is the value b2 - 4ac. It is possible to predict the characteristics of the quadratic equation's roots based on the discriminant value.
Discriminant: D = b2 - 4ac.
If D > 0, then the roots are real and distinct.
If D = 0, then the roots are real and equal.
When D < 0, the roots are complex.

Now observe the formulas for finding the sum and the product of the roots of the given equation.

Sum and Product of Roots of Quadratic Equation 

The sum and product of roots of a quadratic equation can be found using the coefficient of x2, the coefficient of x, and the constant term of the equation ax2 + bx + c = 0. Compute the sum and product of the roots of the equation from the equation. The sum and product of the roots of the quadratic equation ax2 + bx + c = 0 are as follows.

The sum of the roots is α + β = -b / a = - Coefficient of x / Coefficient of x2
The product of the roots is αβ = c / a = Constant term / Coefficient of x2.

What are the Formulas Related to Quadratic Equations 

Formulas help solve quadratic equations faster. There are many formulas related to quadratic equations, and some of the most important ones are mentioned below:

• The standard form of a quadratic equation is ax2 + bx + c = 0
• The discriminant of the quadratic equation is D = b2 - 4ac
• For D > 0, the roots are real and distinct.
• For D = 0, the roots are real and equal.
• For D < 0, the roots are complex.
• The roots of the standard quadratic equation can be found by using the formula:
•         x = (-b ± √(b2 - 4ac)2a.
• The sum of the roots of any quadratic equation is α + β = -b / a.
• The product of the roots of any quadratic equation is αβ = c / a.
• The quadratic equation whose roots are α and β is x2 - (α + β)x + αβ = 0.
• The condition for the quadratic equations a1 x2 + b1 x + c1 = 0 and a2 x2 + b2 x + c2 = 0 to have the same roots is: a1a2 = b1b2 = c1c2. 
• When a > 0, the quadratic expression f(x) = ax2 + bx + c has a minimum value at x = -b/2a.
• When a < 0, the quadratic expression f(x) = ax2 + bx + c has a maximum value at x = -b/2a.
• The domain of any quadratic function is the set of all real numbers.
   

You can find the two roots of the quadratic equations using the four methods listed below:

• Quadratic equation factorization

• Applying the quadratic formula, which we have already seen in the above topics.

• Completing the square

• Graphing the equation

To learn more about the aforementioned techniques, their applications, and their uses, let's take a closer look at them.

How to Solve Quadratic Equations by Factorization

There are several steps involved in factorizing a quadratic equation. We must first split the middle part of the equation into two groups. The split must happen in such a way that their product should be the same as the product of coefficient a and constant c in the equation ax2 + bx + c = 0. Additionally, we can extract the common terms from the available terms to ultimately derive the necessary factors in the manner described below:

• x2 + (a + b) x + ab = 0 
• (x2 + ax) + (bx + ab) = 0 
• x (x + a) + b (x + a) = 0
• (x + a)(x + b) = 0

To better understand the factorization process, consider this example.

• x2 + 5x + 6 = 0 
• x2 + 2x + 3x + 6 = 0 
• x(x + 2) + 3(x + 2) = 0
• (x + 2) (x + 3) = 0

As a result, the quadratic equation's two obtained factors are (x + 2) and (x + 3). Simply set each factor to zero and solve for x to determine its roots. That is, x = -2 and x = -3 because x + 2 = 0 and x + 3 = 0. Therefore, the roots of x2 + 5x + 6 = 0 are x = -2 and x = -3.

Additionally, there is another crucial approach to solving a quadratic equation. Finding the roots of a quadratic equation can also be accomplished by using the method of completing the square.

Completing the square in a quadratic equation involves simplifying and algebraically squaring the equation to find the necessary roots. Examine the quadratic equation ax2 + bx + c = 0, where a ≠ 0. We simplify this equation as follows to find its roots:

ax2 + bx + c = 0.
ax2 + bx = -c

x2 + bx / a = -c / a
By adding a new term (b / 2a)2 to both sides, we can now represent the left-hand side as a perfect square:
x2 + bx / a + (b / 2a)2 = -c/a + (b/2a)2 
(x + b / 2a)2 = -c/a + b2/4a2 
(x + b / 2a)2 = (b2 - 4ac)/4a2

x + b / 2a = +√(b2- 4ac) / 2a 
x = -b / 2a +√(b2 - 4ac) / 2a 
x = [-b ± √(b2 - 4ac)] / 2a
In this case, the quadratic equation's "+" and "-" signs indicate different roots. In most cases, this intricate process is omitted, and the necessary roots are obtained solely by applying the quadratic formula.

How to Graph a Quadratic Equation

By expressing the quadratic equation as a function y = ax2 + bx + c, the graph of the quadratic equation ax2 + bx + c = 0 can be found. Additionally, we can obtain values of y by substituting different values of x, which gives us the required points to plot the graph. To create a parabola, these points can be displayed on the coordinate plane.

The quadratic equation's solutions are the x-values where its graph crosses the x-axis. By setting y = 0 in the function y = ax2 + bx + c and solving for x, these points can be obtained algebraically.
 

Let a1x2 + b1x + c1 = 0 and a2x2 + b2 x + c2 = 0 be two quadratic equations with common roots. To determine the conditions under which these two equations share a root, let's solve them. The two equations are solved for x2 and x.

(x2) (b1 c2 - b2 c1) = (-x) / (a1 c2 - a2 c1) = 1 / (a1 b2 - a2 b1)
x2 = (b1c2 - b2c1) / (a1b2 - a2b1) 
x = (a2c1 - a1c2) / (a1b2 - a2b1)

Therefore, the following condition for the two equations having a common root is obtained by simplifying the two expressions above.
(a1b2 – a2b1) (b1c2 - b2c1) = (a2c1 - a1c2)2

Maximum and Minimum Value of Quadratic Expression

Graphs can be used to represent the maximum and minimum values of a quadratic function in the form f(x) = ax2 + bx + c. If the value of a is positive (a > 0), then the parabola opens upwards and the maximum value of x is at x = -b/2a. It has a maximum value at x = -b/2a for negative values of a (a < 0). And the parabola opens downwards.

The maximum and minimum values are used to determine the range of the unctions:

The range is [f(-b/2a), ∞] for positive values of a (a > 0) 
The range is [(-∞, f(-b/2a))] for negative values of a (a < 0).

Range for a > 0: [ f((- b2a), ∞)]
Range: [(-∞, f (- b2a))] for a < 0.

Keep in mind that the domain of any quadratic function is all real numbers, or (-∞, ∞).
 

This section examines the practical applications of quadratic equations in a variety of domains. Quadratics are used to solve many problems in daily life. For e.g., in a sport like cricket, the concept is used to determine a ball's trajectory. Let us take a look at some other applications of quadratic equations:

• The quadratic equations are used to model the path of objects in projectile motion. For example, to calculate how high a basketball will go.

• In computer graphics and animation quadratic equation is used to model the motion of objects and to create realistic movements in games and cartoons 

• Quadratic equations are used in finance to predict the amount based on variables like interest rates, time, and investment amount.