Quadratic Equations – Solved Problems with Explanations
A quadratic equation is a polynomial equation of degree two. It is one of the most important topics in algebra and is widely used in mathematics, physics, engineering, and competitive examinations.
In this article, we will solve multiple quadratic equations using different methods such as factorization and the quadratic formula, with complete step-by-step explanations.
Standard Form of a Quadratic Equation
A quadratic equation is written in the form:
ax² + bx + c = 0, where a ≠ 0
- a is the coefficient of x²
- b is the coefficient of x
- c is the constant term
Solved Quadratic Equation Problems
Problem 1: Solve x² − 5x + 6 = 0
Solution (Factorization Method):
x² − 5x + 6 = 0
Find two numbers whose product is 6 and sum is −5.
Those numbers are −2 and −3.
(x − 2)(x − 3) = 0
x = 2 or x = 3
Problem 2: Solve x² + 7x + 10 = 0
Solution:
x² + 7x + 10 = 0
(x + 5)(x + 2) = 0
x = −5 or x = −2
Problem 3: Solve 2x² − 3x − 2 = 0
Solution:
2x² − 3x − 2 = 0
Multiply constant and coefficient of x²: 2 × (−2) = −4
Split −3x as −4x + x
2x² − 4x + x − 2 = 0
2x(x − 2) + 1(x − 2) = 0
(x − 2)(2x + 1) = 0
x = 2 or x = −1/2
Problem 4: Solve x² + 4x + 5 = 0
Solution (Quadratic Formula):
Here, a = 1, b = 4, c = 5
Discriminant D = b² − 4ac = 16 − 20 = −4
Since D < 0, the equation has no real roots.
Problem 5: Solve x² − 9 = 0
Solution:
x² − 9 = 0
x² = 9
x = ±3
Problem 6: Solve 3x² + 5x − 2 = 0
Solution:
3x² + 5x − 2 = 0
Split middle term:
3x² + 6x − x − 2 = 0
3x(x + 2) −1(x + 2) = 0
(x + 2)(3x − 1) = 0
x = −2 or x = 1/3
Important Exam Tips
✔ Always write the equation in standard form
✔ Check discriminant to know nature of roots
✔ Use factorization whenever possible
✔ Apply quadratic formula when factorization fails
Common Mistakes to Avoid
- Sign errors while splitting middle term
- Forgetting that a ≠ 0
- Incorrect calculation of discriminant
- Skipping negative square root solution
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