Quadratic Formula: Formula, Examples and Solving Methods
Quadratic Formula is a standard method for solving equations of the form ax² + bx + c = 0, where a ≠ 0. This page explains the formula, discriminant, nature of roots, sign rules and step-by-step quadratic formula examples used in aptitude problems.
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What Is the Quadratic Formula?
For an equation ax² + bx + c = 0, identify a, b and c with their signs before substituting. The expression D = b² - 4ac is called the discriminant. For example, in 2x² - 5x + 2 = 0, a = 2, b = -5 and c = 2, so x = [5 ± √(25 - 16)] / 4 = (5 ± 3) / 4. Therefore, x = 2 or x = 1/2.
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Quadratic Formula and Discriminant
Learn how to use the quadratic formula to solve quadratic equations and apply the discriminant to determine the nature and number of their roots.
Quadratic Formula Formula & Tricks
Important Formulas
Use this formula to find the roots of ax² + bx + c = 0, with a ≠ 0.
The discriminant determines the type and number of real roots.
For roots α and β of ax² + bx + c = 0, their sum is -b/a and their product is c/a.
Quick Tricks
The equation must be in the form ax² + bx + c = 0 before identifying a, b and c. Moving a term changes its sign.
If b is negative, then -b becomes positive. Treat -b as one complete part of the formula.
Calculate b² - 4ac separately. If it is a perfect square, the roots can be found quickly.
Quadratic Formula Concepts
Steps to Apply the Formula
Use these steps: (1) write ax² + bx + c = 0; (2) identify a, b and c with their signs; (3) calculate D = b² - 4ac; (4) substitute in x = (-b ± √D)/2a; and (5) simplify both signs. For x² - 6x + 8 = 0, D = 36 - 32 = 4, so x = (6 ± 2)/2, giving x = 4 and x = 2.
Discriminant and Nature of Roots
If D > 0, the equation has two distinct real roots. If D = 0, it has two equal real roots. If D < 0, it has no real roots; its roots are complex conjugates. When D is a positive perfect square and a, b, c are integers, the roots are rational.
Equal and Complex Roots
When D = 0, both roots are -b/(2a). When D < 0, √D is not real. The roots can be written as (-b ± i√(-D))/(2a), where i² = -1.
Checking Roots Using Their Sum and Product
For ax² + bx + c = 0 with roots α and β, add the two calculated roots and multiply them. Both results must match -b/a and c/a respectively.
Quadratic Formula Data Table
Use the structured data below while following the explanation and examples.
Discriminant and Root Type
The value of the discriminant determines the number and type of roots.
| Condition on D | Root type | Number of real roots |
|---|---|---|
| D > 0 | Two distinct real roots | 2 |
| D = 0 | Two equal real roots | 1 repeated root |
| D < 0 | Complex conjugate roots | 0 real roots |
Note: Here, D = b² - 4ac.
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Quadratic Formula Revision Points
Remember the standard form, formula, discriminant rules and root relationships.
- Standard form: ax² + bx + c = 0, with a ≠ 0.
- Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.
- Discriminant: D = b² - 4ac.
- D > 0 gives two distinct real roots; D = 0 gives equal roots; D < 0 gives no real roots.
- For roots α and β: α + β = -b/a and αβ = c/a.
- Always preserve the signs of b and c while substituting.
Quadratic Formula FAQs
What is the quadratic formula?
For ax² + bx + c = 0, the roots are x = (-b ± √(b² - 4ac)) / 2a, where a ≠ 0.
How do you find the discriminant of a quadratic equation?
Use D = b² - 4ac. For 2x² + 3x - 2 = 0, D = 3² - 4(2)(-2) = 25.
What are the roots of x² - 5x + 6 = 0 using the quadratic formula?
Here a = 1, b = -5 and c = 6. D = 1, so x = (5 ± 1)/2, giving x = 3 and x = 2.
When does a quadratic equation have equal roots?
It has equal roots when b² - 4ac = 0. The repeated root is x = -b/(2a).
What happens when the discriminant is negative?
The equation has no real roots. Its two roots are complex conjugates, written as (-b ± i√(-D))/(2a).
How can the roots obtained from the quadratic formula be checked?
If the roots are α and β, verify that α + β = -b/a and αβ = c/a. For 2x² - 5x + 2 = 0, roots 2 and 1/2 satisfy both relationships.