Quadratic Equations

A quadratic equation is a second-degree polynomial equation written as:
\(ax^2 + bx + c = 0\)

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 \(ax^2  + bx + c = 0\) 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 example., the equation \(x^2 - 3x - 4 = 0 \) 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: 
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\).

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

Here,
a = 1,
b = 2,
c = −15
Using the formula: \(y = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Substituting the values:  \(y = {-2 \pm \sqrt{2^2-4(1)(-15)} \over 2(1)}\)
                              
                                         \(= {-2 \pm \sqrt{4+ 60} \over 2}\)

                                        \(= {-2 \pm \sqrt{64} \over 2}\)
                                       
                                         \( = {{-2 ± 8\over2}}\)

So, y = \({{-2 + 8\over2 }}=  {{6\over2}} = 3\)
        
and,   y \(= {{-2 - 8\over2}} = {{-10\over2 }}= -5\)

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

The quadratic formula, used in solving equations of the form \(ax^2  + bx + c = 0\), 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: \(ax^2  + bx + c = 0, {\text { a ≠ 0}}\).

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

Step 1: \(ax^2 + bx = -c ⇒ x^2 + {{bx\over a}} = {{-c\over a}}\)

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

\(x^2 + {{bx\over a}} + {{({b\over 2a})^2}} = {{-c\over a }}+ {{({b\over 2a})^2}}\)

Now, the left side is a perfect square:

\({(x + {b\over 2x})}^2 = {-c \over a} + {b^2 \over 4a^2} \implies ({x + b \over 2a})^2 = {({b^2 - 4ac \over 4a^2})}\)

Simplifying the equation further, we get:

\({x + {b\over 2a}} = ± \sqrt {b^2 - 4ac \over 2a}\)

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

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. We can determine the type of roots using the discriminant. 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 \(b^2 - 4ac\).

It is possible to predict the characteristics of the quadratic equation's roots based on the discriminant value.
Discriminant: \(D = b^2 - 4ac\).

• 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.

The sum and product of roots of a quadratic equation can be found using the coefficient of \(x^2\), the coefficient of x, and the constant term of the equation \(ax^2 + bx + c = 0\).

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 \(ax^2 + bx + c = 0 \) are as follows.

The sum of the roots is \(\alpha + \beta = \frac{-b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2} \).
The product of the roots is \(\alpha \beta = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2} \).

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 \(ax^2 + bx + c = 0\).
   
• The discriminant of the quadratic equation is \(D = b^2 - 4ac\).
   
  • 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:
  \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\).
   
• The sum of the roots of any quadratic equation is \(α + β = {{-b \over a}}\).
   
• The product of the roots of any quadratic equation is \(αβ = {{c \over a}}\).
   
• The quadratic equation whose roots are α and β is \(x^2 - (α + β)x + αβ = 0\).
   
• The condition for the quadratic equations \(a_1x^2 + b_1x + c_1 = 0 \) and \(a_2x^2 + b_2x + c_2 = 0\), to have the same roots is: \({{{a_1 \over a_2} = {b_1 \over b_2} = {c_1 \over c_2}}}\). 
   
• When a > 0, the quadratic expression \({{f(x) = ax^2 + bx + c}}\) has a minimum value at \({{x = {{-b\over 2a}}}}\).
   
• When a < 0, the quadratic expression \({{f(x) = ax^2 + bx + c}}\) has a maximum value at \({{x = {{-b\over 2a}}}}.\)
   
• 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.

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 \(ax^2 + bx + c = 0\).

Additionally, we can extract the common terms from the available terms to ultimately derive the necessary factors in the manner described below:

• \( x^2 + (a + b)x + ab = 0 \)
• \((x^2 + ax) + (bx + ab) = 0 \)
• \(x(x + a) + b(x + a) = 0 \)
• \((x + a)(x + b) = 0 \)

To better understand the factorization process, consider this example.

• \(x^2 + 5x + 6\)
• \(x^2 + 2x + 3x + 6 = 0  \)
• \(x(x + 2) + 3(x + 2) = 0 \)
• \((x + 2) (x + 3) = 0\)

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 \(x^2 + 5x + 6 = 0\) 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 \(ax^2 + bx + c = 0\), where \(a \ne 0\). We simplify this equation as follows to find its roots:

\(ax^2 + bx + c = 0\)
\(ax^2 + bx = -c\)

\( {{{{x^2+ {{bx \over a}}}} = {{-c \over a}}}}\)

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

\({{x^2 + {{bx \over a}} + {{({b \over 2a})^2}}}} = {{{{-c\over a }}+ {{({b\over 2a})^2}}}}\) 
\({({x + {b\over 2a}})}^2 = {-c \over a} + {b^2 \over 4a^2}\)
\({({{x + b} \over 2a})}^2 = {{{({b^2 - 4ac})} \over 4a^2}}\)
\({x + {b \over 2a}} = {{± \sqrt {b^2 - 4ac} \over 2a}}\)
\(x = {{-b \over 2a }} ± {{\sqrt {(b^2 - 4ac) \over 2a }}} \)
\(x = {{{[-b ± \sqrt {(b^2 - 4ac)]} \over 2a}}} \)

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.

By expressing the quadratic equation as a function \(y = ax^2 + bx + c\), the graph of the quadratic equation \(ax^2 + bx + c = 0\) 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 \(y = ax^2 + bx + c\) and solving for x, these points can be obtained algebraically.

Let \(a_1x^2 + b_1x + c_1 = 0\) and \(a_2x^2 + b_2 x + c_2 = 0\) 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.

\({(x^2)} {(b_1c_2 - b_2 c_2)} = {{{(-x)} \over {(a_1c_2 - a_2c_1)}}} = {{1} \over {(a_1b_2 - a_2b_1)}}\)
\(x^2 ={{ (b_1c_2 - b_2c_1) \over (a_1b_2 - a_2b_1) }} \)
\(x = {{(a_2c_1 - a_1c_2) \over (a_1b_2 - a_2b_1)}}\)

Therefore, the following condition for the two equations having a common root is obtained by simplifying the two expressions above.
\({{(a_1b_2 – a_2b_1) (b_1c_2 - b_2c_1) = (a_2c_1 - a_1c_2)^2}}\)

Graphs can be used to represent the maximum and minimum values of a quadratic function in the form \(f(x) = ax^2 + bx + c\). If the value of a is positive (a > 0), then the parabola opens upwards and the maximum value of x is at \({{x = {{-b\over 2a}}}}\). It has a maximum value at \({x = {-b\over 2a}}\) 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 functions:

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

Range for a > 0: \([f{({-b\over 2a})}, ∞]\)
Range: \([(-∞, f({-b\over 2a}))]\) for a < 0.

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

Mastering quadratic equations becomes easier with the right approach and consistent practice. These tips and tricks help students learn effectively while providing parents with simple, practical ways to guide and support their child’s learning.

• Always start by identifying the value of a, b, and c in the equation \(ax^2 + bx + c = 0\). Knowing these values helps students to choose the best solving method such as factoring, completing the square, or using the quadratic formula.
   
• Understand the pattern, as factoring becomes easy when you recognize these patterns. For example, \(x^2 + 5x + 6 = (x + 2)(x + 3)\).
   
• Memorize the quadratic formula, that is \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) . It helps students to solve the quadratic equation easily. 
   
• Use graph to visualize the quadratic equation, it helps to understand how it behaves. The point where the curve meets the x-axis are the solutions. 
   
• To understand the number of roots, always check \(b^2 - 4ac\). If the value of discriminant is positive it has two real roots, zero it has one real root, and negative it has two complex roots.      

Quadratic equations are not just abstract concepts in mathematics, but they play a crucial role in solving real-world problems across various fields. They are used to modeling situations where the relationship between variables forms a parabolic curve. Here are some practical applications:

• Quadratic equations help determine the path of objects thrown or launched into the air. For instance, they are used to calculate the maximum height a basketball reaches or the distance a cricket ball travels.

• In gaming and animation, quadratics are used to simulate realistic movements of characters and objects. This includes designing jumps, falls, or curved paths of moving elements.

• Quadratic equations are applied to predict profits, losses, or investment growth when factors like interest rates, time, and principal amounts vary.

• Engineers use quadratic equations to design parabolic arches, bridges, and structures, ensuring both strength and aesthetic appeal.

• Quadratics are employed to calculate quantities like acceleration, displacement, and energy in systems where motion follows a curved trajectory.