Quadratic Formula
The quadratic formula provides a universal method for solving any quadratic equation ax² + bx + c = 0. It works for every case — whether the roots are rational, irrational, or complex.
While factorisation and completing the square work for specific types of equations, the quadratic formula guarantees a correct solution when other methods fail.
The formula is derived from completing the square on the general equation, and it expresses the roots directly in terms of the coefficients a, b, and c.
What is Quadratic Formula?
Definition: The quadratic formula is an algebraic expression that gives the roots of any quadratic equation in standard form ax² + bx + c = 0 (where a ≠ 0).
x = (−b ± √(b² − 4ac)) / 2a
Key facts:
- The ± sign indicates two solutions — one using + and another using −.
- The expression b² − 4ac under the square root is the discriminant (D).
- The discriminant determines the nature of roots before calculation.
The formula works for all real values of a, b, and c (including decimals and fractions). It does not require the equation to have factorisable integer roots.
Steps to apply:
- Write the equation in standard form: ax² + bx + c = 0.
- Identify a, b, c (watch the signs carefully).
- Substitute into the formula.
- Simplify the result.
Quadratic Formula Formula
The Quadratic Formula:
x = (−b ± √(b² − 4ac)) / 2a
Where:
- a = coefficient of x² (must not be zero)
- b = coefficient of x
- c = constant term
- b² − 4ac = discriminant (D)
The Two Roots:
- Root 1: x₁ = (−b + √D) / 2a
- Root 2: x₂ = (−b − √D) / 2a
Derivation and Proof
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0.
Proof:
- Start with: ax² + bx + c = 0
- Divide by a (since a ≠ 0): x² + (b/a)x + (c/a) = 0
- Transpose the constant: x² + (b/a)x = −(c/a)
- Add (b/(2a))² = b²/(4a²) to both sides:
x² + (b/a)x + b²/(4a²) = −(c/a) + b²/(4a²) - LHS becomes a perfect square:
(x + b/(2a))² = b²/(4a²) − c/a - Common denominator 4a² on RHS:
(x + b/(2a))² = (b² − 4ac) / (4a²) - Take square root (include ±):
x + b/(2a) = ± √(b² − 4ac) / (2a) - Note: √(4a²) = 2|a| = 2a (the formula holds regardless due to ±).
- Isolate x:
x = (−b ± √(b² − 4ac)) / (2a)
The + sign gives one root and the − sign gives the other. The derivation shows that the quadratic formula is simply the result of completing the square on the most general quadratic equation.
Types and Properties
The quadratic formula produces different results depending on the discriminant D = b² − 4ac:
| Discriminant | Nature of Roots | What Happens |
|---|---|---|
| D > 0, perfect square | Two distinct rational roots | √D is a whole number; roots are integers or fractions |
| D > 0, not perfect square | Two distinct irrational roots | √D is irrational; roots contain surds (e.g., 3 ± √5) |
| D = 0 | Two equal real roots (repeated) | The ± part becomes zero; both roots = −b/(2a) |
| D < 0 | No real roots (complex) | √D involves imaginary unit i; roots are not real |
For CBSE Class 10, students handle cases where D ≥ 0 (real roots). Complex roots are studied in higher classes.
Methods
Steps to solve using the Quadratic Formula:
- Write in standard form: Rearrange so that all terms are on one side: ax² + bx + c = 0.
- Identify coefficients: Find a, b, c. Pay attention to signs. Example: in 3x² − 7x + 2 = 0, b = −7 (not 7).
- Calculate the discriminant: D = b² − 4ac. This reveals the nature of roots.
- Apply the formula: x = (−b ± √D) / (2a).
- Simplify: Reduce fractions and simplify surds.
- Verify: Substitute each root back into the original equation.
Common Pitfalls:
- Forgetting the negative sign with b when substituting into −b.
- Computing (−b)² as −b² instead of b².
- Dividing only the √D part by 2a instead of the entire numerator (−b ± √D).
- Not simplifying the final answer (leaving reducible fractions or unsimplified surds).
Solved Examples
Example 1: Solving with Integer Roots
Problem: Solve x² − 5x + 6 = 0 using the quadratic formula.
Solution:
Given:
- a = 1, b = −5, c = 6
Using x = (−b ± √D) / 2a:
- D = (−5)² − 4(1)(6) = 25 − 24 = 1
- D = 1 > 0 → two distinct real roots
- x = (5 ± √1) / 2 = (5 ± 1) / 2
- x₁ = (5 + 1)/2 = 3
- x₂ = (5 − 1)/2 = 2
Answer: x = 3 or x = 2
Example 2: Solving with Fractional Roots
Problem: Solve 4x² − 12x + 5 = 0 using the quadratic formula.
Solution:
Given:
- a = 4, b = −12, c = 5
Using the formula:
- D = (−12)² − 4(4)(5) = 144 − 80 = 64
- D = 64 (perfect square) → rational roots
- x = (12 ± √64) / 8 = (12 ± 8) / 8
- x₁ = 20/8 = 5/2
- x₂ = 4/8 = 1/2
Answer: x = 5/2 or x = 1/2
Example 3: Solving with Irrational Roots
Problem: Solve x² + 4x + 1 = 0 using the quadratic formula.
Solution:
Given:
- a = 1, b = 4, c = 1
Using the formula:
- D = 4² − 4(1)(1) = 16 − 4 = 12
- D = 12 (not a perfect square) → irrational roots
- x = (−4 ± √12) / 2 = (−4 ± 2√3) / 2
- x₁ = −2 + √3
- x₂ = −2 − √3
Answer: x = −2 + √3 or x = −2 − √3 (approximately −0.268 or −3.732)
Example 4: Equation with Equal Roots
Problem: Solve 9x² − 12x + 4 = 0 using the quadratic formula.
Solution:
Given:
- a = 9, b = −12, c = 4
Using the formula:
- D = (−12)² − 4(9)(4) = 144 − 144 = 0
- D = 0 → two equal roots
- x = 12 / 18 = 2/3
Answer: x = 2/3 (repeated root). Both roots equal 2/3.
Example 5: Equation with No Real Roots
Problem: Determine whether 2x² + 3x + 5 = 0 has real roots.
Solution:
Given:
- a = 2, b = 3, c = 5
Checking discriminant:
- D = 3² − 4(2)(5) = 9 − 40 = −31
- D = −31 < 0 → no real roots
Answer: The equation has no real roots. The parabola y = 2x² + 3x + 5 lies entirely above the x-axis.
Example 6: Equation Requiring Rearrangement
Problem: Solve 3x² = 2x + 5 using the quadratic formula.
Solution:
Rearranging to standard form: 3x² − 2x − 5 = 0
Given:
- a = 3, b = −2, c = −5
Using the formula:
- D = (−2)² − 4(3)(−5) = 4 + 60 = 64
- x = (2 ± √64) / 6 = (2 ± 8) / 6
- x₁ = 10/6 = 5/3
- x₂ = −6/6 = −1
Answer: x = 5/3 or x = −1
Example 7: Word Problem: Ages
Problem: The sum of a father's and son's ages is 45 years. Five years ago, the product of their ages was 124. Find their present ages.
Solution:
Given:
- Sum of ages = 45
- Product of ages 5 years ago = 124
Let son's age = x. Father's age = 45 − x.
Steps:
- Five years ago: son = (x − 5), father = (40 − x)
- (x − 5)(40 − x) = 124
- 40x − x² − 200 + 5x = 124
- −x² + 45x − 200 = 124
- x² − 45x + 324 = 0
- D = 2025 − 1296 = 729 = 27²
- x = (45 ± 27)/2 → x = 36 or x = 9
- If x = 36, father = 9 (impossible — son older than father)
- If x = 9, father = 36 (valid)
Answer: Son is 9 years old, father is 36 years old.
Example 8: Word Problem: Number of Items
Problem: A shopkeeper buys books for Rs 1200. If he bought 5 more books for the same amount, each book would cost Rs 20 less. Find the number of books.
Solution:
Given:
- Total cost = Rs 1200
- 5 more books → Rs 20 less per book
Let number of books = x. Cost per book = 1200/x.
Steps:
- 1200/x − 1200/(x + 5) = 20
- Multiply by x(x + 5): 1200(x + 5) − 1200x = 20x(x + 5)
- 6000 = 20x² + 100x
- 20x² + 100x − 6000 = 0
- Divide by 20: x² + 5x − 300 = 0
- D = 25 + 1200 = 1225 = 35²
- x = (−5 ± 35)/2 → x = 15 or x = −20
- Number of books cannot be negative.
Answer: The shopkeeper bought 15 books.
Example 9: Finding the Value of k for Equal Roots
Problem: Find the value of k for which kx² + 6x + 1 = 0 has equal roots.
Solution:
Given:
- a = k, b = 6, c = 1
For equal roots, D = 0.
Steps:
- D = 6² − 4(k)(1) = 36 − 4k
- Set D = 0: 36 − 4k = 0
- 4k = 36 → k = 9
Verification: 9x² + 6x + 1 = 0 → D = 36 − 36 = 0. Confirmed.
Answer: k = 9
Example 10: Word Problem: Work Rate
Problem: Two pipes together fill a tank in 6 hours. The larger pipe alone fills it 5 hours faster than the smaller pipe. Find the time each pipe takes individually.
Solution:
Given:
- Combined time = 6 hours
- Larger pipe is 5 hours faster than smaller pipe
Let smaller pipe take x hours alone. Larger pipe takes (x − 5) hours.
Steps:
- Combined rate: 1/x + 1/(x − 5) = 1/6
- Multiply by 6x(x − 5): 6(x − 5) + 6x = x(x − 5)
- 6x − 30 + 6x = x² − 5x
- 12x − 30 = x² − 5x
- x² − 17x + 30 = 0
- D = 289 − 120 = 169 = 13²
- x = (17 ± 13)/2 → x = 15 or x = 2
- If x = 2, larger = 2 − 5 = −3 (impossible)
- x = 15. Larger pipe = 15 − 5 = 10 hours.
Answer: Smaller pipe: 15 hours, larger pipe: 10 hours.
Real-World Applications
Projectile Motion:
- Height of a thrown ball at time t: h(t) = −½gt² + ut + h₀.
- Finding when the ball hits the ground (h = 0) requires solving a quadratic equation.
Electronics:
- Finding current or voltage in circuits with resistors, capacitors, and inductors leads to quadratic equations.
Economics:
- Revenue and profit functions are often quadratic.
- Example: R = −2p² + 100p — the quadratic formula finds the price at which revenue equals a target.
Geometry:
- Finding dimensions of shapes (radius, sides, length/breadth) given area and perimeter constraints.
Chemistry:
- Equilibrium constant calculations for weak acids and bases require solving quadratic equations.
Key Points to Remember
- The quadratic formula x = (−b ± √(b² − 4ac)) / 2a gives the roots of any quadratic equation.
- Always write the equation in standard form before identifying a, b, c.
- Be careful with signs — the most common errors come from handling negative b or c.
- The discriminant D = b² − 4ac determines the nature of roots before solving.
- When D is a perfect square, roots are rational (simplifiable to fractions or integers).
- When D is positive but not a perfect square, roots are irrational (expressed in surd form).
- When D = 0, both roots are equal: x = −b/(2a).
- Always simplify the final answer — reduce fractions, simplify surds.
- Verify roots by substituting back into the original equation.
- In word problems, check that solutions make physical sense (reject negative lengths, speeds, counts).
Practice Problems
- Solve using the quadratic formula: x² + 3x − 10 = 0
- Solve using the quadratic formula: 5x² − 8x + 3 = 0
- Solve: 2x² + x − 6 = 0
- Find the roots of 3x² − 2x − 1 = 0 using the quadratic formula.
- Determine whether 4x² − 4x + 1 = 0 has equal, distinct, or no real roots. Then find them.
- Find the value of p for which the equation px² − 14x + 8 = 0 has two equal roots.
- The difference of two numbers is 5, and the difference of their reciprocals is 1/10. Find the numbers using the quadratic formula.
- A rectangular park has a perimeter of 60 m. If its area is 200 m², find the length and breadth using the quadratic formula.
Frequently Asked Questions
Q1. What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the roots of any quadratic equation ax² + bx + c = 0. The ± sign means there are two roots: one using + and another using −.
Q2. When should I use the quadratic formula instead of factorisation?
Use the quadratic formula when the equation cannot be easily factorised, when roots are irrational or involve surds, or when you need a guaranteed method. Factorisation is quicker for equations with integer roots.
Q3. How do I verify my answer from the quadratic formula?
Substitute each root back into the original equation. If the left-hand side equals zero, the answer is correct. Also check: sum of roots = −b/a and product of roots = c/a.
Q4. Why does the quadratic formula have a ± sign?
The ± arises because taking the square root during derivation yields both positive and negative values. This gives two values of x, corresponding to the two roots of the quadratic equation.
Q5. Can the quadratic formula give no solution?
The formula always gives a mathematical result. However, when D = b² − 4ac is negative, √D is not a real number. The equation then has no real roots. The complex roots exist but are beyond Class 10 scope.
Q6. What happens when a = 0 in the quadratic formula?
When a = 0, the equation is linear (bx + c = 0), not quadratic. The formula cannot be applied because division by 2a = 0 is undefined. Solve it as a linear equation: x = −c/b.
Q7. How is the quadratic formula derived?
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. By systematically isolating x through algebraic manipulation, the result is x = (−b ± √(b² − 4ac)) / 2a.
Q8. Is the quadratic formula important for CBSE board exams?
Yes. Questions may ask to solve equations using the formula, derive it, find the nature of roots, or solve word problems. It typically carries 3–5 marks.
Q9. What is the relationship between the quadratic formula and completing the square?
The quadratic formula is the result of completing the square on the general equation ax² + bx + c = 0. Completing the square is the process; the quadratic formula is the final result expressed as a ready-to-use formula.
Q10. Can I use the quadratic formula for equations with decimal coefficients?
Yes. The formula works for any real values of a, b, c, including decimals and fractions. However, it is often easier to multiply the entire equation by a suitable number to eliminate decimals before applying the formula.
Related Topics
- Discriminant of Quadratic Equation
- Completing the Square Method
- Nature of Roots of Quadratic Equation
- Quadratic Equations
- Standard Form of Quadratic Equation
- Solving Quadratic Equations by Factorisation
- Factorisation of Quadratic Expressions
- Sum and Product of Roots
- Word Problems on Quadratic Equations
- Applications of Quadratic Equations
- Roots and Graphs of Quadratic Equations
