Summarize this article:
103 LearnersLast updated on September 25, 2025
Quadratic Equations Formulas
In mathematics, quadratic equations are polynomial equations of degree 2. The general form of a quadratic equation is ax² + bx + c = 0. In this topic, we will learn the formulas related to solving quadratic equations, such as the quadratic formula and methods of factorization.
List of Quadratic Equations Formulas
Quadratic equations are integral to algebra and are used to find the values of unknown variables. Let’s learn the formulas to solve quadratic equations using different methods.
Quadratic Formula
The quadratic formula is used to find the solutions of a quadratic equation ax² + bx + c = 0. It is given by: x = (-b ± √(b² - 4ac)) / (2a)
Factorization Method
The factorization method involves expressing the quadratic equation in a product of linear factors. A quadratic equation ax² + bx + c = 0 can be factored as (px + q)(rx + s) = 0.
Solving these factors gives the roots of the equation.
Completing the Square
Completing the square is another method to solve quadratic equations.
It involves rearranging the equation to form a perfect square trinomial.
The equation ax² + bx + c = 0 can be rewritten as (x + d)² = e, where d and e are constants derived from the coefficients a, b, and c.
Importance of Quadratic Equations Formulas
In mathematics and real-life applications, solving quadratic equations is essential. Here are some important uses of quadratic equations:
- Quadratic equations help in determining the trajectory of objects in physics.
- They are used in calculating areas and optimizing functions in calculus.
- The quadratic formula provides a straightforward way to find roots without factorization.
Tips and Tricks to Memorize Quadratic Equations Formulas
Students may find quadratic formulas challenging, but with some tips, they can master these easily:
- Remember the quadratic formula using the mnemonic: "Negative b, plus or minus the square root, of b squared minus 4ac, all over 2a."
- Practice converting quadratic equations to different forms (standard, vertex, and factored) to get familiar with them.
- Use flashcards to memorize the formulas and practice problems for reinforcement.
Common Mistakes and How to Avoid Them While Using Quadratic Equations Formulas
Students often make errors when solving quadratic equations. Here are some common mistakes and ways to avoid them:
Mistake 1
Forgetting the ± Sign in the Quadratic Formula
When using the quadratic formula, students often forget the ± sign, which leads to missing one of the roots.
Always remember to consider both the positive and negative solutions.
Mistake 2
Incorrect Factorization
Errors occur when students incorrectly factorize the quadratic.
To avoid these, double-check the product and sum match the original equation's coefficients.
Mistake 3
Misplacing Negative Signs
Misplacing negative signs while solving can lead to incorrect solutions.
Pay careful attention to signs, especially when dealing with (b² - 4ac).
Mistake 4
Ignoring Non-Real Solutions
Students sometimes ignore non-real solutions when the discriminant (b² - 4ac) is negative.
Remember that such conditions lead to complex roots.
Mistake 5
Incorrect Completing the Square
Errors in completing the square usually arise from incorrect rearrangement.
Ensure each step converts the equation correctly to a perfect square form.
Examples of Problems Using Quadratic Equations Formulas
Problem 1
Solve the quadratic equation 2x² - 8x + 6 = 0 using the quadratic formula.
The solutions are x = 3 and x = 1.
Explanation
For the equation 2x² - 8x + 6 = 0, a = 2, b = -8, c = 6.
Using the quadratic formula: x = (-(-8) ± √((-8)² - 4*2*6)) / (2*2) x = (8 ± √(64 - 48)) / 4 x = (8 ± √16) / 4 x = (8 ± 4) / 4
The solutions are x = 3 and x = 1.
Problem 2
Solve the quadratic equation x² + 6x + 9 = 0 by factorization.
The solution is x = -3.
Explanation
Factorize the equation x² + 6x + 9 = 0: (x + 3)(x + 3) = 0
The solution is x = -3.
Problem 3
Solve the quadratic equation x² - 4x - 5 = 0 by completing the square.
The solutions are x = 5 and x = -1.
Explanation
Rearrange x² - 4x - 5 = 0: x² - 4x = 5
Complete the square: (x - 2)² = 9 x - 2 = ±3
The solutions are x = 5 and x = -1.
FAQs on Quadratic Equations Formulas
1.What is the quadratic formula?
2.What is the factorization method?
3.How do you complete the square?
4.Why do quadratic equations have two solutions?
5.What happens if the discriminant is negative?
Glossary for Quadratic Equations Formulas
- Quadratic Equation: A polynomial equation of degree 2, typically in the form ax² + bx + c = 0.
- Discriminant: The expression b² - 4ac in the quadratic formula, determining the nature of the roots.
- Roots: The solutions of the quadratic equation, where the graph intersects the x-axis.
- Completing the Square: A method used to solve quadratic equations by forming a perfect square trinomial.
- Factorization: Expressing a quadratic equation as a product of linear factors to find the roots.
Explore More math-formulas
![Important Math Links Icon]()
Previous to Quadratic Equations Formulas
![Important Math Links Icon]()
Next to Quadratic Equations Formulas
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
