solve by completing the square worksheet

Guri Bee

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22-10-2024

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Mastering the Square: A Guide to Completing the Square Worksheets

Completing the square is a powerful algebraic technique used to solve quadratic equations and manipulate expressions. While initially seeming complex, mastering this method unlocks a deeper understanding of quadratic functions and their applications. This article will guide you through the process of completing the square, demystifying the concept using examples from a worksheet format.

What is Completing the Square?

Completing the square involves rewriting a quadratic expression in the form of $(x + h)^2 + k$. This process allows us to isolate the variable 'x' and solve for its value. Here's a step-by-step guide:

Step 1: Standard Form

Ensure your quadratic equation is in standard form: $ax^2 + bx + c = 0$. If the coefficient of $x^2$ is not 1, divide the entire equation by 'a'.

Step 2: Move the Constant

Shift the constant term 'c' to the right side of the equation. This isolates the terms with 'x'.

Step 3: The Key Step: Completing the Square

This is the crucial step. Take half of the coefficient of 'x' (which is 'b'), square it, and add it to both sides of the equation. This ensures we are not changing the original equation's value.

Example:

Let's consider the quadratic equation $x^2 - 6x + 5 = 0$.

• Step 1: Already in standard form.
• Step 2: $x^2 - 6x = -5$
• Step 3: Half of -6 is -3, squaring it gives 9. Add 9 to both sides: $x^2 - 6x + 9 = -5 + 9$

Step 4: Factoring

The left side of the equation can now be factored as a perfect square: $(x - 3)^2 = 4$

Step 5: Solving for x

Take the square root of both sides and solve for 'x':

• $x - 3 = \pm 2$
• $x = 3 \pm 2$
• $x = 5$ or $x = 1$

Completing the Square Worksheet Examples

Here are some examples from a "Solve by Completing the Square" worksheet, illustrating the application of the method:

Example 1:

• Equation: $x^2 + 8x - 9 = 0$
• Solution:
  • Move the constant: $x^2 + 8x = 9$
  • Complete the square: $x^2 + 8x + 16 = 9 + 16$
  • Factor: $(x + 4)^2 = 25$
  • Solve: $x + 4 = \pm 5$
  • Solution: $x = 1$ or $x = -9$

Example 2:

• Equation: $2x^2 - 12x + 10 = 0$
• Solution:
  • Divide by 2: $x^2 - 6x + 5 = 0$
  • Move the constant: $x^2 - 6x = -5$
  • Complete the square: $x^2 - 6x + 9 = -5 + 9$
  • Factor: $(x - 3)^2 = 4$
  • Solve: $x - 3 = \pm 2$
  • Solution: $x = 5$ or $x = 1$

Why Use Completing the Square?

Beyond solving quadratic equations, completing the square has several applications:

• Finding the Vertex of a Parabola: Completing the square transforms the equation of a parabola into vertex form, making it easy to identify the vertex, axis of symmetry, and other key features.
• Deriving the Quadratic Formula: The quadratic formula is derived using the method of completing the square, providing a deeper understanding of its origins and applications.
• Solving Equations in a Standard Form: Completing the square allows for manipulating equations into a standard form, facilitating further analysis and calculations.

Conclusion

Completing the square, while initially challenging, becomes a powerful tool for working with quadratic expressions. This article, utilizing examples from a worksheet format, provided a comprehensive guide to understanding and mastering this crucial algebraic technique. Remember, practice is key! The more you work through these examples and complete your worksheets, the more comfortable and proficient you will become with completing the square.