How to Complete the Square Step by Step with Formula and Examples
The concept of completing the square plays a key role in mathematics, especially when solving quadratic equations that can’t be factored easily. It is a powerful algebraic method with applications in finding roots, graphing parabolas, and understanding vertex form, making it crucial for class tests and competitive exams like JEE and NEET.
What Is Completing the Square?
Completing the square is a way to rewrite a quadratic equation, like ax² + bx + c, into a perfect square form such as a(x + h)² + k. This method helps us solve quadratic equations, find roots, convert to vertex form, and graph parabolas. You’ll find completing the square used in areas such as quadratic equations, vertex form conversion, and parabola graphing.
Key Formula for Completing the Square
Here’s the standard formula: ax² + bx + c = a(x + \(\dfrac{b}{2a}\))² - a\(\left(\dfrac{b^2 - 4ac}{4a^2}\right)\).
How to Complete the Square: Step by Step
To complete the square in any quadratic equation ax² + bx + c = 0:
- Move the constant 'c' to the other side.
- Divide all terms by 'a' (if a ≠ 1).
- Add (b/2a)² to both sides.
- Rewrite the left side as a squared binomial.
- Solve for x (take square roots and simplify).
Step-by-Step Illustration
- Start with the equation: \(x^2 + 6x + 5 = 0\)
- Move 5 to the RHS: \(x^2 + 6x = -5\)
- Add \((6/2)^2 = 9\) to both sides: \(x^2 + 6x + 9 = -5 + 9\)
- Simplify: \(x^2 + 6x + 9 = 4\)
- Write as a square: \((x+3)^2 = 4\)
- Take square roots: \(x+3 = \pm2\)
- Final Answer: \(x = -3 \pm 2\) ⇒ \(x = -1\), \(x = -5\)
Common Mistakes & Tips
- Forgetting to add (b/2a)² to both sides, leading to an unbalanced equation.
- Incorrectly calculating b/2a (not dividing b by 2 AND a).
- Not dividing all terms by 'a' in cases where a ≠ 1.
- Missing the ± when taking square roots.
- Skipping steps in exam solutions—always write each step for full marks!
Try These Yourself
- Solve \(x^2 + 8x + 7 = 0\) using completing the square.
- Find the vertex form of \(2x^2 + 4x + 1\).
- Convert \(x^2 - 10x + 9\) to its vertex form.
- Solve \(3x^2 + 12x - 27 = 0\) by completing the square.
Relation to Other Concepts
The idea of completing the square is closely tied to quadratics, factoring quadratics, and the quadratic formula. In fact, the quadratic formula is derived using the method of completing the square. Mastering this helps students understand vertex form and how to graph any parabola with confidence.
Classroom Tip
A quick way to remember completing the square: Always find b/2a, square it, and add it to both sides! Vedantu’s teachers often recommend writing out every step, including balancing both sides, to avoid mistakes in exams.
Wrapping It All Up
We have explored completing the square—from definition and core formula to stepwise solutions, tips, and how it links to other algebraic methods. With regular practice and learning at Vedantu, you can quickly solve even the trickiest quadratic equations and score better in Maths!
For more about quadratic equations and related topics, check out:
FAQs on Completing the Square for Quadratic Equations
1. What is completing the square in maths?
Completing the square is a method used to rewrite a quadratic expression in the form a(x − h)² + k. It converts a quadratic like ax² + bx + c into vertex form so you can easily find the vertex, minimum/maximum value, or solve equations.
For example:
- x² + 6x + 5
- = (x² + 6x + 9) − 9 + 5
- = (x + 3)² − 4
2. How do you complete the square step by step?
To complete the square, you add and subtract the square of half the coefficient of x. Follow these steps for x² + bx + c:
- 1. Take half of b.
- 2. Square it.
- 3. Add and subtract this value inside the expression.
- Half of 8 is 4.
- 4² = 16.
- x² + 8x + 16 − 16 + 3
- = (x + 4)² − 13
3. What is the formula for completing the square?
The completing the square formula for x² + bx + c is (x + b/2)² − (b/2)² + c. This comes from adding and subtracting the square of half the coefficient of x.
For a general quadratic ax² + bx + c:
- Factor out a from the first two terms.
- Complete the square inside the bracket.
- Rewrite in the form a(x − h)² + k.
4. How do you complete the square when a is not 1?
When the coefficient of x² is not 1, first factor it out before completing the square. For example, solve 2x² + 8x + 3:
- Factor 2 from first two terms: 2(x² + 4x) + 3
- Half of 4 is 2, and 2² = 4
- 2(x² + 4x + 4 − 4) + 3
- = 2(x + 2)² − 8 + 3
- = 2(x + 2)² − 5
5. Why do we complete the square?
We complete the square to rewrite a quadratic in vertex form so we can easily find its turning point, minimum or maximum value, and solve quadratic equations. It is especially useful for:
- Finding the vertex (h, k)
- Deriving the quadratic formula
- Solving equations without factoring
- Graphing parabolas
6. How do you solve a quadratic equation by completing the square?
To solve a quadratic by completing the square, rewrite the equation so one side is a perfect square and then take the square root. Example: Solve x² + 6x + 5 = 0.
- Move constant: x² + 6x = −5
- Half of 6 is 3, and 3² = 9
- x² + 6x + 9 = 4
- (x + 3)² = 4
- x + 3 = ±2
- x = −1 or −5
7. How does completing the square help find the vertex of a parabola?
Completing the square rewrites a quadratic in the form a(x − h)² + k, where (h, k) is the vertex. In this form:
- The vertex is (h, k)
- If a > 0, the parabola opens upward
- If a < 0, it opens downward
8. What is the difference between factoring and completing the square?
Factoring rewrites a quadratic as a product, while completing the square rewrites it as a perfect square expression. For example:
- Factoring: x² + 5x + 6 = (x + 2)(x + 3)
- Completing the square: x² + 5x + 6 = (x + 5/2)² − 1/4
9. What are common mistakes when completing the square?
The most common mistake in completing the square is forgetting to add and subtract the same value. Watch out for:
- Not halving the coefficient of x correctly
- Forgetting to square the half value
- Missing the factor a when a ≠ 1
- Sign errors when moving constants
10. How is completing the square used to derive the quadratic formula?
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. The steps are:
- Divide by a.
- Move the constant term.
- Complete the square.
- Take the square root and simplify.
