Quadratic equations might seem intimidating at first, but once you understand the different solving methods, they become much easier to handle. Whether you're a Class 10 student or just brushing up on algebra, this guide will walk you through five key methods to solve any quadratic equation, complete with examples.
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial in the form:
ax² + bx + c = 0
Where:
a, b, c are real numbers (a ≠ 0).
x represents the variable.
These equations appear in real-world scenarios like calculating projectile motion, optimizing profit in business, or determining dimensions in geometry.
Related: How to crack International Mathematical Olympiad?
5 Methods to Solve Quadratic Equations
1. Factoring Method (Best for simple, factorable equations)
When to use: If the quadratic can be broken into two binomials.
Steps:
Write the equation in standard form: ax² + bx + c = 0.
Find two numbers that:
Multiply to a × c.
Add to b.
Split the middle term and factor by grouping.
Set each factor to zero and solve for x.
Example: Solve x² + 5x + 6 = 0
Numbers: 2 and 3 (since 2 × 3 = 6 and 2 + 3 = 5).
Factored form: (x + 2)(x + 3) = 0
Solutions: x = -2, x = -3
2. Completing the Square (Useful when factoring is difficult)
When to use: For equations that don’t factor easily.
Steps:
Move the constant term (c) to the other side.
Divide all terms by a (if a ≠ 1).
Take half of b, square it, and add to both sides.
Rewrite the left side as a perfect square.
Solve for x.
Example: Solve x² + 6x + 5 = 0
Rewrite: x² + 6x = -5
Complete the square: x² + 6x + 9 = 4 (since (6/2)² = 9)
Perfect square: (x + 3)² = 4
Solutions: x = -1, x = -5
3. Quadratic Formula (Works for any quadratic equation)
The universal method for solving quadratics:
Formula:
Steps:
Identify a, b, c from the equation.
Plug into the formula.
Simplify under the square root (discriminant).
Solve for x.
Example: Solve 2x² + 4x – 6 = 0
a = 2, b = 4, c = -6
Discriminant: √(16 + 48) = √64 = 8
Solutions: x = 1, x = -3
4. Splitting the Middle Term (Alternative to factoring)
When to use: For equations like ax² + bx + c where a ≠ 1.
Steps:
Multiply a × c.
Find two numbers that multiply to a × c and add to b.
Split the middle term and factor.
Example: Solve 6x² + 11x + 4 = 0
a × c = 24 → Numbers: 8 and 3 (8 + 3 = 11)
Rewrite: 6x² + 8x + 3x + 4 = 0
Factor: (2x + 1)(3x + 4) = 0
Solutions: x = -½, x = -4/3
Related: Is computer science the same as software engineering math?
5. Graphical Method (Visual approach for real-world applications)
When to use: To estimate real roots visually.
Steps:
Plot the quadratic function y = ax² + bx + c.
The x-intercepts (where y = 0) are the solutions.
Example: Graph y = x² – 4 → Roots at x = 2, x = -2.
FAQs on Quadratic Equations
Q: Which method is easiest?
A: Factoring (if the equation is simple). Otherwise, the quadratic formula always works.
Q: What if the discriminant is negative?
A: The equation has no real roots (only complex solutions).
Q: How is this useful in real life?
A: Used in physics (projectile motion), engineering (structural design), and finance (profit optimization).
Q: What’s the biggest mistake students make?
A: Forgetting to check all solutions or misapplying the quadratic formula.
Final Tips for Solving Quadratics
✔ Memorize the quadratic formula – it’s a lifesaver!
✔ Practice factoring – it’s faster when it works.
✔ Check your answers by plugging them back into the equation.
Now you’re ready to tackle any quadratic equation! Try solving these:
x² – 9 = 0
2x² + 5x – 3 = 0
Drop your answers in the comments!
Matti Karttunen
Quadratic equations might seem intimidating at first, but once you understand the different solving methods, they become much easier to handle. Whether you're a Class 10 student or just brushing up on algebra, this guide will walk you through five key methods to solve any quadratic equation, complete with examples.
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial in the form:
ax² + bx + c = 0
Where:
a, b, c are real numbers (a ≠ 0).
x represents the variable.
These equations appear in real-world scenarios like calculating projectile motion, optimizing profit in business, or determining dimensions in geometry.
Related: How to crack International Mathematical Olympiad?
5 Methods to Solve Quadratic Equations
1. Factoring Method (Best for simple, factorable equations)
When to use: If the quadratic can be broken into two binomials.
Steps:
Write the equation in standard form: ax² + bx + c = 0.
Find two numbers that:
Multiply to a × c.
Add to b.
Split the middle term and factor by grouping.
Set each factor to zero and solve for x.
Example: Solve x² + 5x + 6 = 0
Numbers: 2 and 3 (since 2 × 3 = 6 and 2 + 3 = 5).
Factored form: (x + 2)(x + 3) = 0
Solutions: x = -2, x = -3
2. Completing the Square (Useful when factoring is difficult)
When to use: For equations that don’t factor easily.
Steps:
Move the constant term (c) to the other side.
Divide all terms by a (if a ≠ 1).
Take half of b, square it, and add to both sides.
Rewrite the left side as a perfect square.
Solve for x.
Example: Solve x² + 6x + 5 = 0
Rewrite: x² + 6x = -5
Complete the square: x² + 6x + 9 = 4 (since (6/2)² = 9)
Perfect square: (x + 3)² = 4
Solutions: x = -1, x = -5
3. Quadratic Formula (Works for any quadratic equation)
The universal method for solving quadratics:
Formula:
Steps:
Identify a, b, c from the equation.
Plug into the formula.
Simplify under the square root (discriminant).
Solve for x.
Example: Solve 2x² + 4x – 6 = 0
a = 2, b = 4, c = -6
Discriminant: √(16 + 48) = √64 = 8
Solutions: x = 1, x = -3
4. Splitting the Middle Term (Alternative to factoring)
When to use: For equations like ax² + bx + c where a ≠ 1.
Steps:
Multiply a × c.
Find two numbers that multiply to a × c and add to b.
Split the middle term and factor.
Example: Solve 6x² + 11x + 4 = 0
a × c = 24 → Numbers: 8 and 3 (8 + 3 = 11)
Rewrite: 6x² + 8x + 3x + 4 = 0
Factor: (2x + 1)(3x + 4) = 0
Solutions: x = -½, x = -4/3
Related: Is computer science the same as software engineering math?
5. Graphical Method (Visual approach for real-world applications)
When to use: To estimate real roots visually.
Steps:
Plot the quadratic function y = ax² + bx + c.
The x-intercepts (where y = 0) are the solutions.
Example: Graph y = x² – 4 → Roots at x = 2, x = -2.
FAQs on Quadratic Equations
Q: Which method is easiest?
A: Factoring (if the equation is simple). Otherwise, the quadratic formula always works.
Q: What if the discriminant is negative?
A: The equation has no real roots (only complex solutions).
Q: How is this useful in real life?
A: Used in physics (projectile motion), engineering (structural design), and finance (profit optimization).
Q: What’s the biggest mistake students make?
A: Forgetting to check all solutions or misapplying the quadratic formula.
Final Tips for Solving Quadratics
✔ Memorize the quadratic formula – it’s a lifesaver!
✔ Practice factoring – it’s faster when it works.
✔ Check your answers by plugging them back into the equation.
Now you’re ready to tackle any quadratic equation! Try solving these:
x² – 9 = 0
2x² + 5x – 3 = 0
Drop your answers in the comments!