Among the most significant concepts of mathematics are quadratic equations. They are the basis of higher mathematics on algebra, coordinate geometry and calculus. Be it board exams, competitive tests or simply refreshing your basic, quadratic equations are a requirement.
We will discuss quadratic equations step by step in this guide, including quadratic equations examples, tricks, important questions and real-life application as well. At the end, you will not just know how to solve them but you will also come to appreciate how extensively they have been used.
What is a Quadratic Equation?
The quadratic equation is a 2 degree equation. It is usually contrived in the following form:
ax^2 + bx + c = 0
where:
– a, b, c are real numbers (with a ≠ 0),
– x is the variable.
Examples:
– x^2 – 5x + 6 = 0
– 2x^2 + 3x – 2 = 0
In this case, the maximum power of x is 2 thus they are quadratic equations.
Standard Form of Quadratic Equation.
All quadratic equations may be written as:
ax^2 + bx + c = 0
a: second-degree term coefficient of x 2
b: co-efficient of x (linear term)
c: constant term
Example: In the equation: 2x 2 +7x -15=0 we can find:
a = 2,
b = 7,
c = -15
Step 1: Solving Quadratic Equations.
Quadratic equations can be solved in four major ways:
1. Factorization Method
This consists of23 splitting the middle term into two factors.
Example: Solve x^2 – 5x + 6 = 0.
Step 1: Multiply a and c: 1 × 6 = 6.
Step 2: Find factors of 6 that add to -5: (-2, -3).
Step 3: Rewrite: x^2 – 2x – 3x + 6 = 0.
Step 4: Group terms: x(x – 2) – 3(x – 2) = 0.
Step 5: Factorize: (x – 2)(x – 3) = 0.
Step 6: Solutions: x = 2, 3.
It is among the easiest quadratic equation tricks in case the numbers are conveniently factorable.
2. Completing the Square
We interchange words just to form a square of perfection.
Example: Solve x^2 + 6x + 5 = 0.
Step 1:: Write x^2 + 6x and leave space.
Step 2:: Half of 6 = 3 → 3^2 = 9.
Step 3:: Add & subtract 9: (x^2 + 6x + 9) – 9 + 5 = 0.
Step 4:: Simplify: (x + 3)^2 – 4 = 0.
Step 5:: (x + 3)^2 = 4.
Step 6:: x + 3 = ±2.
Step 7:: Solutions: x = -1, -5.
3. Quadratic Formula
The most universal method:
x = (-b ± √(b^2 – 4ac)) / 2a
The part under the square root, b^2 – 4ac, is called the discriminant (D).
– If D > 0: Two distinct real roots.
– If D = 0: One real root (repeated).
– If D < 0: No real roots (complex solutions).
Example: Solve 2x^2 + 3x – 2 = 0.
– a = 2, b = 3, c = -2.
– D = b^2 – 4ac = 9 – (4×2×-2) = 9 + 16 = 25.
– x = (-3 ± √25) / 4.
– x = (-3 ± 5) / 4.
– Roots: x = 0.5, -2.
4. Graphical Method
y = ax 2 + bx + c is a parabola.
- Where the parabola crosses the x-axis, those x-values are the real roots.
- If it crosses at two points, there are two real roots.
- If it just touches the x-axis at one point, there is one repeated real root.
- If it does not touch the x-axis at all, there are no real roots.”
It is a visual and practical method of knowing the solutions.
Step 2: Quadratic Equations Important Questions
In order to master the topic, practice is needed. Important questions you should attempt here are a few quadratic equations:
- Solve: x^2 – 7x + 12 = 0.
- Solve using quadratic formula: 3x^2 – 5x + 2 = 0.
- Find the roots of x^2 + 4x + 4 = 0.
- If one root of 2x^2 + kx + 3 = 0 is -1, find k.
- The sum and product of roots are 8 and 15. Form the quadratic equation.
Step 3: Real World Applications of Quadratic Equations.
Quadratic equations are not mere mathematical problems; it is all around us. Applications of quadratic equations are:
- Physics: Motion equations like s = ut + ½at^2 are quadratic.
- Architecture: Parabolic arches in bridges.
- Finance: Profit and revenue maximization problems.
- Engineering: Projectile motion and design of satellite dishes.
- Nature: The Path of a ball or comet follows a parabola.
Fast solving of Quadratic Equations Tricks.
The following are some quadratic equations tricks that you can apply in the exams:
- Sum and Product of Roots:
– Sum of roots = -b/a.
– Product of roots = c/a.
– Assists in either creating or checking quadratic equations. - Shortcut to Perfect Squares:
If x^2 + 2ax + a^2 = 0, then roots are -a. - Coefficient Comparison:
And knowing one of the roots, find the other by the product of roots.. - Graph Sketching:
Approximate root positions can be obtained by quick drawings of parabolas.
Common Mistakes to Avoid
– Forgetting to check if a ≠ 0.
– Negative sign omission in using the quadratic formula.
– Suppose all the quadratics have two real roots (discriminant cases).
What is the Importance of Quadratic Equations?
Quadratic equations form the backbone of algebra and are used in:
– Competitive exams like JEE, NEET, and Olympiads.
– Advanced math topics (differentiation, integration).
– Physical, economic, and engineering problems.
By practicing important questions under quadratic equations, you can be sure that when utilizing the questions on exams and being logical, you will not lose confidence.
Final Thoughts
Learning quadratic equations step by step do not only enable students to score well in exams, it also can ensure that they can use mathematics in practice. This can be mastered with a regular practice of examples of quadratic equations, tricks in solving equations, and not to mention trying important questions.
Quadratic equations are used all over, whether in physics to study the velocity of some moving body or in business to calculate the profit in a particular case or in exams to solve some kind of puzzle. And accept them, do them regularly and watch mathematics become more rational and enjoyable.
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