LSC2: ALGEBRA 2

ALGEBRA 2

Quadratic Expressions

A quadratic expression is an expression with the variable with the highest power of 2. The word quadratic is derived from the word quad which means square.

The expression should have the power of two and not higher or lower.

Graphically a quadratic expression describes the path followed by a parabola, and it helps in finding the height and time of flight of a rocket.

An expression of the form ax² + bx + c, where a ≠ 0 is called a quadratic expression. In other words, any expression with the variables of the highest exponent or the expression’s degree as 2 is a quadratic expression The standard form of a quadratic expression looks like this

Here are some examples of expressions:

Look at example 4, here a = 0. Therefore, it is not a quadratic expression. In fact, this is a linear expression. The remaining expressions are quadratic expression examples.

The standard form of the standard expression in variable x is ax² + bx + c. But always remember that ‘a’ is a non-zero value and sometimes a quadratic expression is not written in its standard form.

Listed below are a few important properties to keep in mind while identifying quadratic expressions.

• The expression is usually written in terms of x, y, z, or w. In the alphabets, the letters, in the end, are written for variables whereas the letters, in the beginning, are used for numbers.
• The variable ‘a’ in a quadratic expression raised to the power of 2 cannot be zero. If a = 0 then x² will be multiplied by zero and therefore, it would not be a quadratic expression anymore. Variable b or c in the standard form can be 0 but ‘a’ cannot.
• In a quadratic expression, it is not necessary that all the terms are positive. The standard form is written in a positive form. However, if the numbers are negative the term will also be negative.
• The terms in a quadratic expression are usually written with the power of 2 first, the power of 1 next, and the number in the end.

Ways of Writing a Quadratic Expression

Writing any expression or equation into a quadratic standard form, we need to follow these three methods.

• Move an expression by -1 if the equation starts with a negative value.
• Expand the terms and simply.
• Multiply the factors.

Mentioned below are a few examples:

The graph of the quadratic expression ax² + bx + c = 0 can be obtained by representing the quadratic expression as a function y = ax² + bx + c.

Further on solving and substituting values for x, we can obtain values of y, we can obtain numerous points. These points can be presented in the coordinate axis to obtain a parabola-shaped graph for the quadratic expression.

The point where the graph cuts the horizontal x-axis is the solution of the quadratic expression. These points can also be algebraically obtained by equalizing the y value to 0 in the function y = ax² + bx + c and solving for x. This is how the quadratic expression is represented on a graph. This curve is called a parabola.

The factorizing of a quadratic expression is helpful for splitting it into two simpler expressions, which on multiplying, gives back the original quadratic expression.

The aim of factorization is the break down the expression of higher degrees into expressions of lower degrees. Here the quadratic expression is of the second degree and is split into two simple linear expressions.

The process of factorization of a quadratic equation includes the first step of splitting the x term, such that the product of the coefficients of the x-term is equal to the constant term. Further, the new expression after splitting the x term has four terms in the quadratic expression.

Here some of the terms are taken commonly to obtain the final factors of the quadratic expression. Let us understand the process of factorizing a quadratic expression through an example.

Example 1: 4x – 12x²

Given any quadratic expression, first, check for common factors, i.e. 4x and 12x²

We can observe that 4x is a common factor. Let’s take that common factor from the quadratic expression.

4x – 12x² = 4x(1 – 3x)

Thus, by simplifying the quadratic expression, we get the expression, 4x – 12x².

Solving a quadratic expression is possible if we are able to convert it into a quadratic equation by equalizing it to zero. The values of the variable x which satisfy the quadratic expression and equalize it to zero are called the zeros of the equation. Some of the expressions cannot be easily solved by the method of factorizing. The quadratic formula is here to help. The quadratic formula is also known as “Quadranator.” Quadranator alone is enough to solve all quadratic expression problems. The quadratic expressions formula is as follows.

Discriminant

The discriminant is an important part of the quadratic expression formula. The value of the discriminant is (b² – 4ac). This is called a discriminant because it discriminates the zeroes of the quadratic expression based on its sign. Many an instance we wish to know more about the zeros of the equation, before calculating the roots of the expression. Here the discriminant value is useful. Based on the values of the discriminant the nature of the zeros of the equation or the roots of the equation can be predicted.

The nature of the roots of the quadratic expression based on the discriminant value is as follows.

• When the quadratic expression has two real and distinct roots: b²– 4ac > 0
• When the quadratic expression has equal roots: b²– 4ac = 0
• When the quadratic expression does not have any roots or has imaginary roots: b²– 4ac < 0