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The Quadratic Formula
For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square.
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The Quadratic Formula
What is it?
For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square.
In simple words
- For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a.
- THE FORMULA: Numerator is -b, not b.
- EXAMPLE NUMBERS: x2 - 5x + 6 = 0: a=1, b=-5, c=6.
- Worked-method reminder: x = 62 = 3 or x = 42 = 2
Important terms
THE QUADRATIC FORMULA: For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square.
THE FORMULA: Numerator is -b, not b. Denominator is 2a, not a. The +/- gives two roots (unless D=0). Keep the fraction bar over the whole numerator.
EXAMPLE NUMBERS: x2 - 5x + 6 = 0: a=1, b=-5, c=6. -b=5. D=25-24=1. x = 5 +/- 12 so x=3 or x=2. Same as factoring.
What you should learn
- Explain the quadratic formula using the correct definitions, conditions, and notation.
- Apply x = -b +/- sqrt(D)2a, D = b^2 - 4ac correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- a, b, c from standard form
- sqrt only of D
- Two values if D>0
THE FORMULA
Numerator is -b, not b. Denominator is 2a, not a. The +/- gives two roots (unless D=0). Keep the fraction bar over the whole numerator.
EXAMPLE NUMBERS
x2 - 5x + 6 = 0: a=1, b=-5, c=6. -b=5. D=25-24=1. x = 5 +/- 12 so x=3 or x=2. Same as factoring.
Recommended for Class 10
Full school-level explanation
For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square.
Plug in, then simplify:
WHAT — THE QUADRATIC FORMULA
For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simplify the square root if it is a perfect square. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
The Quadratic Formula matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE FORMULA, then EXAMPLE NUMBERS. Worked: solve x^2 - 5x + 6 = 0 by formula. a = 1, b = -5, c = 6 D = (-5)^2 - 4(1)(6) D = 25 - 24 D = 1 x = 5 +/- 12 x = 62 = 3 or x = 42 = 2 Finish by checking signs, restrictions, units, and whether the answer matches the question.
THE FORMULA
Numerator is -b, not b. Denominator is 2a, not a. The +/- gives two roots (unless D=0). Keep the fraction bar over the whole numerator.
Remember
- -b and 2a
Examples
- a, b, c from standard form
- sqrt only of D
- Two values if D>0
EXAMPLE NUMBERS
x2 - 5x + 6 = 0: a=1, b=-5, c=6. -b=5. D=25-24=1. x = 5 +/- 12 so x=3 or x=2. Same as factoring.
Remember
- matches factoring
Examples
- D=1
- x=5+12=3
- x=5-12=2
Remember
- x = -b +/- sqrt(D)2a
- D = b2 - 4ac
- x = 3 or x = 2
- (-b)^2 vs -b^2
- -b and 2a
- matches factoring
- x = -b +/- sqrt(D)2a
- D = b2 - 4ac
How the ideas connect
Worked: solve x^2 - 5x + 6 = 0 by formula. a = 1, b = -5, c = 6 D = (-5)^2 - 4(1)(6) D = 25 - 24 D = 1 x = 5 +/- 12 x = 62 = 3 or x = 42 = 2
Deeper understanding
Go beyond memorising The Quadratic Formula. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve x^2 - 5x + 6 = 0 by formula. a = 1, b = -5, c = 6 D = (-5)^2 - 4(1)(6) D = 25 - 24 D = 1 x = 5 +/- 12 x = 62 = 3 or x = 42 = 2
Common mistakes
Exam focus
Core Idea
THE FORMULA
EXAMPLE NUMBERS
Worked Method
Exam focus
Quick revision
- x = -b +/- sqrt(D)2a
- D = b2 - 4ac
- x = 3 or x = 2
- (-b)^2 vs -b^2
- THE QUADRATIC FORMULA: For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant
- THE FORMULA: Numerator is -b, not b. Denominator is 2a, not a. The +/- gives two roots (unless D=0). Keep the fra
(-5)^2 is 25, not -25. Square the whole of b, including its sign.
For long answers on The Quadratic Formula: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- x = -b +/- sqrt(D)2a
- D = b2 - 4ac
- x = 3 or x = 2
- (-b)^2 vs -b^2
Important formulas
- x = -b +/- sqrt(D)2a
- D = b2 - 4ac
- x = 3 or x = 2
Common exam traps
- (-5)^2 is 25, not -25. Square the whole of b, including its sign.
- Do not treat THE FORMULA and EXAMPLE NUMBERS as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
(-5)^2 is 25, not -25. Square the whole of b, including its sign.
- Mistake
Do not treat THE FORMULA and EXAMPLE NUMBERS as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
(-5)^2 is 25, not -25. Square the whole of b, including its sign.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
x = -b +/- sqrt(D)2a
D = b2 - 4ac
x = 3 or x = 2
(-b)^2 vs -b^2
For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simp
x = -b +/- sqrt(D)2a
- Definition
- For ax2 + bx + c = 0, x = -b +/- sqrt(b2 - 4ac)2a. The piece b^2 - 4ac is the discriminant D. Substitute a, b, c carefully, including minus signs. Simp
- Formula
- x = -b +/- sqrt(D)2a
- Example
- a, b, c from standard form
- Critical warning
- (-5)^2 is 25, not -25. Square the whole of b, including its sign.
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Exam Practice
Define
Q1. Define The Quadratic Formula.
Explain
Q2. Explain: THE FORMULA.
Example
Q3. Give one clear example of The Quadratic Formula and say why it fits.
Mistake
Q4. State one common mistake students make about The Quadratic Formula.
Exam note
Q5. Write a short exam note on The Quadratic Formula: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.