How to Complete the Square: Step by Step
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: Steps, Formula & Problems
1. What are the steps to complete the square in a quadratic equation?
Completing the square involves rewriting a quadratic equation in the form ax² + bx + c = 0 into a perfect square trinomial. Here's how:
- Move the constant term (c) to the right side of the equation: ax² + bx = -c
- If a ≠ 1, divide all terms by a to get x² + (b/a)x = -c/a
- Add (b/2a)² to both sides of the equation to complete the square: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
- Factor the left side as a perfect square: (x + b/2a)² = -c/a + (b/2a)²
- Solve for x by taking the square root of both sides and simplifying.
2. What does completing the square mean in mathematics?
Completing the square is a method used to solve quadratic equations and to rewrite them in vertex form. It transforms a quadratic expression of the form ax² + bx + c into a perfect square trinomial, making it easier to find the roots (solutions) or to determine the parabola's vertex and axis of symmetry.
3. What is the formula used in completing the square?
There isn't one single formula, but the process relies on the key step of adding (b/2a)² to both sides of the equation ax² + bx + c = 0. This term creates a perfect square trinomial on one side, allowing factorization into the form a(x + h)² + k = 0 (vertex form).
4. How is completing the square used to find the roots of a quadratic equation?
Completing the square transforms the quadratic equation into a form where you can easily isolate x. Once the equation is in the form (x + h)² = k, taking the square root of both sides allows you to solve for the two possible values of x, which are the roots (or solutions) of the quadratic equation.
5. Why is completing the square a useful method for solving quadratic equations?
Completing the square is beneficial when factoring is difficult or impossible. It provides a systematic way to solve any quadratic equation, unlike factoring, which only works for certain types. It's also crucial for converting to vertex form, enabling easy identification of a parabola's vertex and axis of symmetry. This is important for graphing and analyzing quadratic functions.
6. What are some common mistakes to avoid when completing the square?
Common errors include: incorrectly calculating (b/2a)², forgetting to add (b/2a)² to both sides of the equation, errors in factoring the perfect square trinomial, and arithmetic mistakes during simplification.
7. How does completing the square relate to the vertex form of a quadratic?
Completing the square directly leads to the vertex form of a quadratic equation, which is written as a(x - h)² + k = 0, where (h, k) represents the vertex of the parabola. The process of completing the square reveals the values of h and k directly.
8. Can completing the square be used for all quadratic equations?
Yes, completing the square is a universally applicable method for solving all quadratic equations, regardless of whether they are factorable or not. However, for simple equations, factoring may be quicker.
9. What is the relationship between completing the square and the quadratic formula?
The quadratic formula is actually derived by applying the method of completing the square to the general quadratic equation, ax² + bx + c = 0. The formula provides a shortcut to find the roots without going through the step-by-step completing the square process.
10. How do I use completing the square to find the vertex of a parabola?
By completing the square and rewriting the quadratic in vertex form, a(x - h)² + k = 0, you immediately identify the coordinates of the vertex as (h, k). The value of h represents the x-coordinate and k represents the y-coordinate of the vertex.
11. Provide an example of completing the square with a ≠ 1.
Let's solve 2x² + 8x - 10 = 0 using completing the square:
1. 2x² + 8x = 10
2. x² + 4x = 5 (divide by 2)
3. x² + 4x + 4 = 9 (add 4 to both sides)
4. (x + 2)² = 9 (factor)
5. x + 2 = ±3 (take square root)
6. x = 1 or x = -5 (solve for x)
12. When should I choose completing the square over other methods for solving quadratic equations?
Use completing the square when factoring isn't straightforward or when you need the equation in vertex form to easily find the parabola's vertex, axis of symmetry, or to analyze its characteristics. Although the quadratic formula always works, completing the square offers insights into the structure and properties of the quadratic function.
