Completing the Square Method for Solving Quadratic Equations
Completing the square is an elegant algebraic technique that transforms a quadratic equation into a form where the solution can be obtained by simply taking a square root. Unlike factorisation, which only works neatly when the discriminant is a perfect square, completing the square works for every quadratic equation — it always produces the exact roots, whether rational or irrational. This method is historically important as it is the very technique used to derive the quadratic formula itself. For CBSE Class 10 students, completing the square serves a dual purpose: it is a standalone method for solving quadratic equations, and understanding it provides deep insight into why the quadratic formula works. The process involves converting ax2 + bx + c = 0 into the form (x + p)2 = q, from which x can be found by taking the square root of both sides. While the algebraic manipulations require careful handling, the method is systematic and reliable. In this topic, we will learn the step-by-step process, understand the geometric motivation behind the name, and work through a thorough collection of examples to develop mastery.
What is Completing the Square?
Completing the square is a method of rewriting a quadratic expression ax2 + bx + c in the form a(x + d)2 + e, where d and e are constants. When applied to the equation ax2 + bx + c = 0, this transformation allows us to isolate (x + d)2 on one side and solve by taking the square root.
The name 'completing the square' comes from the geometric interpretation. Consider the expression x2 + 6x. If we think of x2 as a square of side x and 6x as a rectangle of dimensions 6 by x, we can split the rectangle into two strips of 3 by x and attach them to two sides of the square. This forms an incomplete larger square of side (x + 3). The missing corner piece is a 3 × 3 square of area 9. Adding this area 'completes the square': x2 + 6x + 9 = (x + 3)2.
Algebraically, the key identity used is:
x2 + 2kx + k2 = (x + k)2
To 'complete the square' for x2 + bx, we take half of b (i.e., b/2), square it (getting b2/4), and add this to the expression. This converts x2 + bx into a perfect square minus the adjustment: x2 + bx + (b/2)2 = (x + b/2)2.
Completing the Square Method for Solving Quadratic Equations Formula
Completing the Square — General Process:
For the equation ax2 + bx + c = 0:
Step 1: Divide through by a (making the leading coefficient 1):
x2 + (b/a)x + c/a = 0
Step 2: Move the constant to the right side:
x2 + (b/a)x = -c/a
Step 3: Add (b/(2a))2 to both sides:
x2 + (b/a)x + (b/(2a))2 = -c/a + b2/(4a2)
Step 4: Write the left side as a perfect square:
(x + b/(2a))2 = (b2 - 4ac) / (4a2)
(x + b/(2a))2 = (b2 - 4ac) / (4a2)
Step 5: Take the square root of both sides and solve for x.
Note: The expression b2 - 4ac is the discriminant D. For real roots, D must be non-negative.
Derivation and Proof
Let us derive the quadratic formula using completing the square, showing how this method leads to the general solution.
Start: ax2 + bx + c = 0 (a ≠ 0)
Step 1: Divide by a:
x2 + (b/a)x + c/a = 0
Step 2: Rearrange:
x2 + (b/a)x = -c/a
Step 3: Half of b/a is b/(2a). Square it: b2/(4a2). Add to both sides:
x2 + (b/a)x + b2/(4a2) = -c/a + b2/(4a2)
Step 4: Left side is a perfect square:
(x + b/(2a))2 = b2/(4a2) - c/a
Step 5: Combine the right side with common denominator 4a2:
(x + b/(2a))2 = (b2 - 4ac) / (4a2)
Step 6: Take the square root of both sides:
x + b/(2a) = ± √(b2 - 4ac) / (2a)
Step 7: Solve for x:
x = -b/(2a) ± √(b2 - 4ac) / (2a)
x = (-b ± √(b2 - 4ac)) / (2a)
This is the quadratic formula. Thus, completing the square is the method that underpins the quadratic formula itself. Every time you use the quadratic formula, you are implicitly using the result of completing the square.
Methods
Practical Steps for Solving by Completing the Square:
For equations with a = 1 (coefficient of x2 is 1):
Example: x2 + 6x + 2 = 0
- Move the constant: x2 + 6x = -2
- Half of 6 is 3. Square: 9. Add to both sides: x2 + 6x + 9 = -2 + 9 = 7
- Write as perfect square: (x + 3)2 = 7
- Take square root: x + 3 = ± √7
- Solve: x = -3 + √7 or x = -3 - √7
For equations with a ≠ 1:
Example: 2x2 + 8x + 3 = 0
- Divide by 2: x2 + 4x + 3/2 = 0
- Move constant: x2 + 4x = -3/2
- Half of 4 is 2. Square: 4. Add: x2 + 4x + 4 = -3/2 + 4 = 5/2
- (x + 2)2 = 5/2
- x + 2 = ± √(5/2) = ± √10/2
- x = -2 ± √10/2
Common Mistakes to Avoid:
- Forgetting to divide by a when a ≠ 1 before completing the square.
- Adding (b/2)2 to only one side of the equation (must add to both sides).
- Miscalculating half of b/a, especially with fractions or negatives.
- Forgetting the ± when taking the square root.
- Not simplifying the final radical answer.
Solved Examples
Example 1: Simple Case with a = 1 (Integer Roots)
Problem: Solve x2 + 10x + 21 = 0 by completing the square.
Solution:
Step 1: Move constant: x2 + 10x = -21
Step 2: Half of 10 is 5. Square: 25. Add to both sides:
x2 + 10x + 25 = -21 + 25 = 4
Step 3: (x + 5)2 = 4
Step 4: x + 5 = ± 2
Step 5: x = -5 + 2 = -3 or x = -5 - 2 = -7
Verification: (-3)2 + 10(-3) + 21 = 9 - 30 + 21 = 0 ✓
(-7)2 + 10(-7) + 21 = 49 - 70 + 21 = 0 ✓
Answer: x = -3 or x = -7.
Example 2: Case with a = 1 (Irrational Roots)
Problem: Solve x2 - 4x + 1 = 0 by completing the square.
Solution:
Step 1: x2 - 4x = -1
Step 2: Half of -4 is -2. Square: 4. Add to both sides:
x2 - 4x + 4 = -1 + 4 = 3
Step 3: (x - 2)2 = 3
Step 4: x - 2 = ± √3
Step 5: x = 2 + √3 or x = 2 - √3
Approximate values: x ≈ 3.732 or x ≈ 0.268
Answer: x = 2 + √3 or x = 2 - √3.
Example 3: Case with a = 2
Problem: Solve 2x2 - 12x + 7 = 0 by completing the square.
Solution:
Step 1: Divide by 2: x2 - 6x + 7/2 = 0
Step 2: x2 - 6x = -7/2
Step 3: Half of -6 is -3. Square: 9. Add to both sides:
x2 - 6x + 9 = -7/2 + 9 = -7/2 + 18/2 = 11/2
Step 4: (x - 3)2 = 11/2
Step 5: x - 3 = ± √(11/2) = ± √22/2
Step 6: x = 3 ± √22/2 = (6 ± √22)/2
Answer: x = (6 + √22)/2 or x = (6 - √22)/2.
Example 4: Equation with Equal Roots
Problem: Solve x2 + 8x + 16 = 0 by completing the square.
Solution:
Step 1: x2 + 8x = -16
Step 2: Half of 8 is 4. Square: 16. Add to both sides:
x2 + 8x + 16 = -16 + 16 = 0
Step 3: (x + 4)2 = 0
Step 4: x + 4 = 0
Step 5: x = -4
The equation has a repeated root x = -4. This occurs because D = 64 - 64 = 0.
Answer: x = -4 (equal roots).
Example 5: Equation with No Real Roots
Problem: Attempt to solve x2 + 2x + 5 = 0 by completing the square.
Solution:
Step 1: x2 + 2x = -5
Step 2: Half of 2 is 1. Square: 1. Add to both sides:
x2 + 2x + 1 = -5 + 1 = -4
Step 3: (x + 1)2 = -4
Step 4: The right side is negative. No real number has a negative square.
Therefore, the equation has no real roots. D = 4 - 20 = -16 < 0 confirms this.
Answer: No real roots exist for this equation.
Example 6: Equation with a = 3 and Odd b
Problem: Solve 3x2 + 5x - 2 = 0 by completing the square.
Solution:
Step 1: Divide by 3: x2 + (5/3)x - 2/3 = 0
Step 2: x2 + (5/3)x = 2/3
Step 3: Half of 5/3 is 5/6. Square: 25/36. Add to both sides:
x2 + (5/3)x + 25/36 = 2/3 + 25/36 = 24/36 + 25/36 = 49/36
Step 4: (x + 5/6)2 = 49/36
Step 5: x + 5/6 = ± 7/6
Step 6: x = -5/6 + 7/6 = 2/6 = 1/3 or x = -5/6 - 7/6 = -12/6 = -2
Verification: 3(1/3)2 + 5(1/3) - 2 = 1/3 + 5/3 - 2 = 6/3 - 2 = 0 ✓
3(-2)2 + 5(-2) - 2 = 12 - 10 - 2 = 0 ✓
Answer: x = 1/3 or x = -2.
Example 7: Equation Requiring Rearrangement First
Problem: Solve 4x2 = 8x + 3 by completing the square.
Solution:
Step 1: Rearrange: 4x2 - 8x - 3 = 0
Step 2: Divide by 4: x2 - 2x - 3/4 = 0
Step 3: x2 - 2x = 3/4
Step 4: Half of -2 is -1. Square: 1. Add to both sides:
x2 - 2x + 1 = 3/4 + 1 = 7/4
Step 5: (x - 1)2 = 7/4
Step 6: x - 1 = ± √7/2
Step 7: x = 1 ± √7/2 = (2 ± √7)/2
Answer: x = (2 + √7)/2 or x = (2 - √7)/2.
Example 8: Completing the Square with a Negative Leading Coefficient
Problem: Solve -x2 + 6x - 5 = 0 by completing the square.
Solution:
Step 1: Multiply by -1: x2 - 6x + 5 = 0
Step 2: x2 - 6x = -5
Step 3: Half of -6 is -3. Square: 9. Add to both sides:
x2 - 6x + 9 = -5 + 9 = 4
Step 4: (x - 3)2 = 4
Step 5: x - 3 = ± 2
Step 6: x = 5 or x = 1
Verification: -(5)2 + 6(5) - 5 = -25 + 30 - 5 = 0 ✓
-(1)2 + 6(1) - 5 = -1 + 6 - 5 = 0 ✓
Answer: x = 5 or x = 1.
Example 9: NCERT-Style Problem
Problem: Find the roots of 5x2 - 6x - 2 = 0 by completing the square.
Solution:
Step 1: Divide by 5: x2 - (6/5)x - 2/5 = 0
Step 2: x2 - (6/5)x = 2/5
Step 3: Half of -6/5 is -3/5. Square: 9/25. Add to both sides:
x2 - (6/5)x + 9/25 = 2/5 + 9/25 = 10/25 + 9/25 = 19/25
Step 4: (x - 3/5)2 = 19/25
Step 5: x - 3/5 = ± √19/5
Step 6: x = (3 ± √19)/5
Answer: x = (3 + √19)/5 or x = (3 - √19)/5.
Example 10: Application: Finding Vertex Form
Problem: Express y = 2x2 - 8x + 11 in the form y = a(x - h)2 + k and find the vertex.
Solution:
Step 1: Factor out 2 from the first two terms:
y = 2(x2 - 4x) + 11
Step 2: Complete the square inside the bracket:
Half of -4 is -2. Square: 4.
y = 2(x2 - 4x + 4 - 4) + 11
y = 2((x - 2)2 - 4) + 11
y = 2(x - 2)2 - 8 + 11
y = 2(x - 2)2 + 3
Step 3: The vertex form is y = 2(x - 2)2 + 3, so the vertex is (2, 3).
Answer: y = 2(x - 2)2 + 3. Vertex = (2, 3). The parabola opens upward (since a = 2 > 0) with minimum value y = 3 at x = 2.
Real-World Applications
Completing the square has applications far beyond solving equations:
Deriving the Quadratic Formula: The quadratic formula is directly derived using completing the square. Understanding this derivation gives deeper insight into why the formula works.
Finding the Vertex of a Parabola: Converting y = ax2 + bx + c to vertex form y = a(x - h)2 + k uses completing the square. The vertex (h, k) gives the maximum or minimum point of the parabola.
Integration in Calculus: Completing the square is used to evaluate certain integrals involving quadratic expressions in the denominator, a technique crucial in higher mathematics.
Conic Sections: Equations of circles, ellipses, and hyperbolas often require completing the square to convert from general form to standard form, revealing the centre and radii.
Optimisation: Expressing a quadratic function in completed square form immediately reveals its maximum or minimum value, useful in business, physics, and engineering applications.
Signal Processing: Completing the square appears in electrical engineering when analysing circuits with quadratic transfer functions.
Key Points to Remember
- Completing the square transforms ax2 + bx + c into a(x + d)2 + e.
- The key step: add (half of the coefficient of x)2 to both sides.
- Always divide by a first when a ≠ 1.
- The method works for ALL quadratic equations — rational, irrational, or no real roots.
- When (x + d)2 equals a negative number, there are no real roots.
- Completing the square is the method used to derive the quadratic formula.
- It converts y = ax2 + bx + c into vertex form y = a(x - h)2 + k.
- Remember to add (b/(2a))2 to BOTH sides of the equation.
- Always include ± when taking the square root.
- Simplify radicals in the final answer (e.g., √50 = 5√2).
Practice Problems
- Solve by completing the square: x² + 4x - 5 = 0.
- Solve by completing the square: x² - 8x + 7 = 0.
- Solve by completing the square: 2x² + 10x + 3 = 0.
- Solve by completing the square: 3x² - 6x + 1 = 0.
- Show that x² + 3x + 4 = 0 has no real roots by completing the square.
- Convert y = x² - 6x + 13 into vertex form using completing the square.
Frequently Asked Questions
Q1. What is completing the square?
Completing the square is a method of rewriting a quadratic expression ax² + bx + c in the form a(x + d)² + e by adding and subtracting a specific constant. This transforms the quadratic equation into a form solvable by taking a square root.
Q2. Why is it called 'completing the square'?
The name comes from a geometric interpretation. The expression x² + bx represents the area of an incomplete square. Adding (b/2)² 'completes' it into a perfect square of side (x + b/2), hence the name.
Q3. When should I use completing the square instead of factorisation?
Use completing the square when the quadratic does not factorise neatly (when D is not a perfect square), or when you need the exact irrational roots. Factorisation is quicker when it works, but completing the square always works.
Q4. What is the first step in completing the square?
If the coefficient of x² (that is, a) is not 1, divide the entire equation by a first. Then move the constant term to the right side. After that, add (half of the coefficient of x)² to both sides.
Q5. What if the right side becomes negative after completing the square?
If (x + d)² = negative number, then the equation has no real roots, because no real number squared gives a negative result. The discriminant in this case is negative.
Q6. How does completing the square relate to the quadratic formula?
The quadratic formula x = (-b ± √(b² - 4ac))/(2a) is derived by applying completing the square to the general equation ax² + bx + c = 0. So the quadratic formula is essentially a pre-computed result of completing the square.
Q7. Can completing the square be used for expressions, not just equations?
Yes. Completing the square can rewrite any quadratic expression (not just equations set equal to zero) in vertex form. This is useful for finding minimum/maximum values and for graphing parabolas.
Q8. Is completing the square tested in CBSE Class 10 board exams?
Yes. CBSE board exams may specifically ask students to solve equations by completing the square. It also helps understand the derivation of the quadratic formula, which is part of the NCERT syllabus.
Q9. What is vertex form and how is it related to completing the square?
Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. Converting from standard form to vertex form is done by completing the square. The vertex gives the maximum or minimum value of the quadratic function.
Q10. What common mistakes should I avoid?
The most common mistakes are: (1) forgetting to divide by a when a ≠ 1, (2) adding (b/2)² to only one side, (3) computing half of b incorrectly when b is a fraction, and (4) forgetting ± when taking the square root.
Related Topics
- Quadratic Formula
- Solving Quadratic Equations by Factorisation
- Quadratic Equations
- Discriminant of Quadratic Equation
- Standard Form of Quadratic Equation
- Factorisation of Quadratic Expressions
- Nature of Roots of Quadratic Equation
- Sum and Product of Roots
- Word Problems on Quadratic Equations
- Applications of Quadratic Equations
- Roots and Graphs of Quadratic Equations
