Quadratic equations are the polynomial equations of degree 2 in one variable of form f(x) = ax² + bx + c where a, b, c, ∈ R and a ≠ 0. It is the standard form of a quadratic equation where ‘a’ is considered the leading coefficient and ‘c’ is called the actual term of f (x). The roots of the quadratic equation (,) are the values of x which also satisfy the quadratic equation.
The quadratic equation will always have exactly two roots. The roots may be either real or imaginary in nature.
Definition of Quadratic Equations:
A quadratic equation is a second-degree equation, meaning it requires at least one squared expression. ax² + bx + c = 0 is the generic type, with a, b, and c being numerical coefficients and x being an unknown variable.
What are Quadratic Equations used for?
Quadratic equations are frequently used in daily activities, such as when measuring regions, calculating a product’s revenue, or calculating an object’s speed. Equations with at least one squared variable are categorized as quadratic equations, with the most common form being ax² + bx + c = 0. The letter X denotes an unknown, with the coefficients a, b, and c denoting known numbers, and the letter denoting that it is not equal to zero. Below mentioned are a few areas where quadratic equations are used:
- Calculating room areas
- Figuring a profit
- Using quadratics in athletics
- Finding speed
Quadratic Equations in Three Forms:
Here are the three primary forms of a quadratic equation and how they should be written:
- Standard form: y = ax² + bx + c where the alphabets a,b, and c are just numbers
- In the factored form: y = (ax + c)(bx + d) here again the alphabets a,b,c, and d are only numbers
- Vertex form: y = a(x + b)² + c here again the alphabets a, b, and c are just numbers
1. Quadratic Equation in the standard form is shown as, ax² + bx + c = 0
2. Quadratic Equations can be factored in as well.
3. Quadratic Formula is stated as, x = −b ± √(b² – 4ac)/2a
4. When the Discriminant (b² − 4ac):
- Is positive, there are only 2 real solutions
- Is zero, there is only one real solution
- Is negative, there are only 2 complex solutions
Different Ways to Solve Quadratic Equations:
Solving quadratic equations can be challenging, but there are a few different approaches we can take depending on the form of quadratic we’re dealing with. Factoring, using square roots, completing the square, and using the quadratic formula are the four methods for solving a quadratic equation.
Solving Quadratic Equations with Factorization:
- Step 1: Find any two numbers that multiply to give ac as a resultant answer (in other words a times c), and add to give b as the final answer.
- Step 2: Rewrite the middle with those respective numbers:
- Step 3: Factor the first two numbers and last two terms separately and not together.
How to Solve a Quadratic Equation?
(i) First we need to simplify the given equation in the general form of the quadratic equation ax² + bx + c = 0, then follow step 2
(ii) We need to factorize the left side of the quadratic equation first before the right side
(iii) Now express each of the two factors equals 0 and solve them separately
(iv)The two solutions are recognized as the roots of the given quadratic equation.
Quadratic equations might get confusing for students, but after consistent practice, students can master the working of quadratic equations. These will be much easier to understand and grasp when practiced with the help of extra online courses and worksheets. Quadratic equations will help students enhance their logical and reasoning skills along with accuracy. Cuemath is an excellent online learning platform that enables students to practise a variety of problems based on Quadratic equations. The Cuemath website helps with tips and techniques that will surely help you excel in these topics.