Completing the Square Method

Oct 13, 2025 by miodeya.com 149 views

Introduction

The quadratic formula is a fundamental concept in algebra, used to solve quadratic equations of the form ax^2 + bx + c = 0. One of the methods to derive the quadratic formula is by completing the square. This method involves manipulating the quadratic equation to create a perfect square trinomial on one side of the equation. In this article, we will explore the expression that can be added to both sides of the equation to create a perfect square trinomial.

Completing the Square Method

The completing the square method involves adding and subtracting a constant term to the quadratic equation to create a perfect square trinomial. The general form of a quadratic equation is ax^2 + bx + c = 0. To complete the square, we need to add and subtract (b/2a)^2 to the equation.

Adding and Subtracting (b/2a)^2

When we add and subtract (b/2a)^2 to the equation, we get:

ax^2 + bx + (b/2a)^2 - (b/2a)^2 + c = 0

We can rewrite this equation as:

ax^2 + bx + (b/2a)^2 = (b/2a)^2 - c

Creating a Perfect Square Trinomial

The left-hand side of the equation is a perfect square trinomial, which can be written in the form:

(a(x + b/2a)^2

To create a perfect square trinomial, we need to add (b/2a)^2 to both sides of the equation.

Expression to be Added

The expression that can be added to both sides of the equation to create a perfect square trinomial is:

(b/2a)^2

This expression is added to both sides of the equation to create a perfect square trinomial on the left-hand side.

Deriving the Quadratic Formula

Using the expression (b/2a)^2, we can derive the quadratic formula by completing the square. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

To derive this formula, we need to add (b/2a)^2 to both sides of the equation and then simplify the resulting expression.

Simplifying the Expression

When we add (b/2a)^2 to both sides of the equation, we get:

ax^2 + bx + (b/2a)^2 = (b/2a)^2 - c

We can rewrite this equation as:

a(x + b/2a)^2 = (b/2a)^2 - c

Taking the square root of both sides, we get:

x + b/2a = ±√((b/2a)^2 - c/a)

Simplifying the expression, we get:

x = (-b ± √(b^2 - 4ac)) / 2a

Conclusion

In conclusion, the expression that can be added to both sides of the equation to create a perfect square trinomial is (b/2a)^2. This expression is used to derive the quadratic formula by completing the square. The quadratic formula is a fundamental concept in algebra, used to solve quadratic equations of the form ax^2 + bx + c = 0.

Perfect Square Trinomial Expression

The perfect square trinomial expression is:

(a(x + b/2a)^2

This expression is created by adding (b/2a)^2 to both sides of the equation.

Quadratic Formula Derivation

The quadratic formula is derived by completing the square and adding (b/2a)^2 to both sides of the equation. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

Final Thoughts

In this article, we explored the expression that can be added to both sides of the equation to create a perfect square trinomial. We also derived the quadratic formula by completing the square and adding (b/2a)^2 to both sides of the equation. The quadratic formula is a fundamental concept in algebra, used to solve quadratic equations of the form ax^2 + bx + c = 0.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Quadratic Equations" by Math Open Reference

Glossary

Quadratic Formula: A formula used to solve quadratic equations of the form ax^2 + bx + c = 0.
Perfect Square Trinomial: A trinomial that can be written in the form (x + a)^2.
Completing the Square: A method used to manipulate a quadratic equation to create a perfect square trinomial on one side of the equation.