Quadratic Expressions

The word "quadratic" originates from the Latin word "quadratus," meaning "square," indicating that the highest power of a quadratic expression is two. Mathematically, a quadratic expression is written as ax2 + bx + c, where a  0. Here, a and b are the coefficients, and c is the constant. 
Some varied examples of quadratic expressions are:
Standard quadratic form: 3x2 + 2x + 1 
Without linear term (b = 0): 2x2 + 5 
Without constant term(c = 0): − 3x2 − 9x
Only quadratic term (b = 0, c = 0): −x2
With fractions or decimals: 12x2-3x+54, 0.25x2 + 1.5x-2
 

Expressions and equations both contain numbers and variables, but they are different from each other in the following ways:
 

Some key properties of quadratic expressions are:

• In algebra, letters like w, x, y, and z are often used to represent variables, while a, b and c are used for constants and coefficients in quadratic expressions.

• In the standard quadratic expression ax2 + bx + c. The coefficient a can never be zero. If a = 0 then the expression becomes linear. However, b and c can be zero.

• The terms in a quadratic expression can be positive or negative depending on the numbers involved. Not all terms are required to be positive

• In the standard form of a quadratic expression, the terms are arranged in descending order of power. The first term is square, then the linear term, and then the constant term.

How to Graph Quadratic Expressions?

To graph a quadratic expression in the form ax2 + bx + c, we use the function: y = ax2 + bx + c
This helps us connect each value of x to a corresponding y value. Then, pick different values for x, positive and negative. Substitute them into equations and calculate the values of y. This gives us a set of (x, y) coordinate points. Upon plotting these points on a graph, we get a parabola.
 

Factorization simplifies a quadratic expression into two linear expressions. To do so, we start by splitting the middle term such that the sum of the two terms equals the middle term and their product is the product of the first and last terms. Then we group the terms and factor out common terms. This completes the factorization of quadratic equations.

For example: Factorize x2 + 5x + 6

Step 1:  Multiply the coefficient of x2 (which is 1) and the constant term (6): 1 × 6 = 6

Step 2: Find two numbers that add up to 5 and multiply to give 6.
Those numbers are 2 and 3.

Step 3: Split the middle term using 2 and 3: x2 + 2x + 3x + 6
 

Step 4: Group the terms and factor: (x2 + 2x) + (3x + 6) = x(x + 2) + 3(x + 2)
 

Step 5: Take out the common binomial: (x + 2) (x + 3)
So, x2 + 5x + 6 = (x + 2) (x + 3)

What is the formula for Quadratic Expressions?

To solve a quadratic expression, we first change it to an equation by equating it to zero ax2 + bx + c = 0. 
The values of x in this equation are called the zeroes or roots of the quadratic equation. 
The quadratic formula x=-b  b2 - 4ac2a gives the values of x for the equation mentioned above.

What is Discriminant in Quadratic Expressions?

The discriminant tells us the nature of roots based on its value. By calculating the discriminant, you can predict how many solutions the equation has, whether they are real or complex, and if they are equal or different.
It can be found using the formula:
Discriminant (D) = b2 - 4ac
Here, a,b, and c are the coefficients from the standard form of a quadratic expression.
 

Quadratic expressions are used in various real-life areas, including:

Calculating projectile motion in physics and sports

The quadratic expression h(t) = -16t2 + vt + h0 calculates the maximum height or time to hit the ground for any object following a parabolic trajectory.

Finding dimensions and solving area problems in architecture and design

Quadratic expressions help find dimensions and optimize space. For instance, the area expression for a garden having length x meters and width x + 3 meters would be A = x(x + 3) = x2 + 3x.

Creating realistic paths in animation

In animation, quadratic expressions are used to determine the u-position over time when a character moves along a curved path.

Building arches, bridges, and curved roads

Civil engineers use quadratic expressions to design curved structures like bridges, parabolic roads, and arches.

Designing a satellite dish

Satellite dishes are paraboloid-shaped to focus signals at the receiver. This parabolic shape is modeled using a quadratic expression to make sure that all incoming signals are reflected onto the focal point.