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Quadratic Formula Derivation

Step-by-step derivation of x from the quadratic equation ax² + bx + c = 0.

📐 Making x the Subject: ax² + bx + c = 0

This is the quadratic formula. Below is the complete step-by-step derivation.

Step 1: Start with the equation

ax² + bx + c = 0

Step 2: Move c to the other side

ax² + bx = -c

Step 3: Divide both sides by a

x² + (b/a)x = -c/a

Step 4: Complete the square

Take half of b/a, square it, and add to both sides:

Half of b/a is b/(2a)
Square it: (b/(2a))² = b²/(4a²)

x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²)

Step 5: Factor the left side

(x + b/(2a))² = b²/(4a²) - c/a

Step 6: Put the right side over a common denominator

(x + b/(2a))² = (b² - 4ac) / (4a²)

Step 7: Take the square root of both sides

x + b/(2a) = ± √[(b² - 4ac) / (4a²)]

x + b/(2a) = ± √(b² - 4ac) / (2a)

Step 8: Solve for x

x = -b/(2a) ± √(b² - 4ac) / (2a)

Step 9: Combine over a common denominator

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Final Answer

x = (-b ± √(b² - 4ac)) / (2a)

Summary Table

Step	Operation	Result
1	Start	ax² + bx + c = 0
2	Move c	ax² + bx = -c
3	Divide by a	x² + (b/a)x = -c/a
4	Complete the square	x² + (b/a)x + b²/(4a²) = b²/(4a²) - c/a
5	Factor	(x + b/(2a))² = (b² - 4ac)/(4a²)
6	Square root	x + b/(2a) = ± √(b² - 4ac)/(2a)
7	Solve for x	x = (-b ± √(b² - 4ac)) / (2a)

Notes

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Key Points

• The expression b² - 4ac is called the discriminant
• If b² - 4ac > 0 → two real solutions
• If b² - 4ac = 0 → one real solution
• If b² - 4ac → no real solutions (complex solutions)

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📐 Summary: The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is derived by completing the square and provides the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\).