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Transforming a quadratic expression

Transforming a quadratic expression into a different form, specifically by completing the square, is a crucial technique in algebra. This transformation simplifies the analysis of the quadratic expression. In its standard form, \( ax^2 + bx + c \), the expression can be difficult to analyze because it has two terms that depend on \( x \). By completing the square, we transform the expression into a form that clearly shows its characteristics, such as the vertex and axis of symmetry.

Why Transform Quadratic Expressions?

The primary goal of this transformation is to make it easier to understand the behavior of the quadratic function. It helps in finding the range, maximum and minimum values, and understanding the conditions for the output to increase or decrease.

Steps to Complete the Square

  1. Start with the General Form: The quadratic expression is given by: \( ax^2 + bx + c \)

  2. Factor Out \( a \) from the First Two Terms (if \( a \neq 1 \)): Factor \( a \) out of the first two terms to get: \( a(x^2 + \frac{b}{a}x) + c \)

  3. Add and Subtract \(\left(\frac{b}{2a}\right)^2\): To form a perfect square trinomial, add and subtract \(\left(\frac{b}{2a}\right)^2\): \( a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \)

  4. Rearrange the Expression: Rearrange to form the perfect square: \( a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2\right) - a\left(\frac{b}{2a}\right)^2 + c \)

  5. Form the Perfect Square: The expression inside the brackets is now a perfect square: \( a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c \)

  6. Simplify the Expression: Simplify to get the final form: \( a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c \)

  7. One last thing: Add the last two terms and incorporate the this term called as discriminant \( \Delta = b^2 - 4ac \): \( a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a} \)

Completed square Form

The completed square form of the quadratic expression is: \( a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a} \)

This form makes it easier to analyze the quadratic function's properties, including its range, graph, maximum/minimum values. It also provides insights into the nature of its roots through the discriminant term \( \Delta\).

Example Let us apply it to the expression \( 2x^2 - 5x - 1 \). Completing the square for the quadratic expression \( 2x^2 - 5x - 1 \) involves several steps to transform it into a more analyzable form. This method helps in understanding the function's behavior more clearly.

Steps to Complete the Square for \( 2x^2 - 5x - 1 \)

  1. Start with the Original Expression: \( 2x^2 - 5x - 1 \)

  2. Factor Out the Coefficient of \( x^2 \): Since the coefficient of \( x^2 \) is 2, we factor it out from the first two terms: \( 2(x^2 - \frac{5}{2}x) - 1 \)

  3. Add and Subtract the Square of Half the Coefficient of \( x \): Add and subtract \(\left(\frac{-5}{2 \times 2}\right)^2\) inside the bracket: \( 2\left(x^2 - \frac{5}{2}x + \left(\frac{-5}{4}\right)^2 - \left(\frac{-5}{4}\right)^2\right) - 1 \)

  4. Rearrange the Expression: Keep the perfect square inside the bracket and move the subtracted term outside: \( 2\left(x^2 - \frac{5}{2}x + \left(\frac{-5}{4}\right)^2\right) - 2\left(\frac{-5}{4}\right)^2 - 1 \)

  5. Form the Perfect Square: The expression inside the bracket is now a perfect square: \( 2\left(x - \frac{5}{4}\right)^2 - 2\left(\frac{-5}{4}\right)^2 - 1 \)

  6. Simplify the Expression: Simplify the terms outside the perfect square: \( 2\left(x - \frac{5}{4}\right)^2 - \frac{25}{8} - 1 =\) \( 2\left(x - \frac{5}{4}\right)^2 - \frac{33}{8} \)

Final Transformed Expression

The completed square form of the expression \( 2x^2 - 5x - 1 \) is: \( 2\left(x - \frac{5}{4}\right)^2 - \frac{33}{8} \)

Comparing the two forms of the quadratic expression \( 2x^2 - 5x - 1 \) highlights the advantages of the completed square form, especially in terms of simplifying the expression's dependence on \( x \).

Original Form: \( 2x^2 - 5x - 1 \)

  • In this standard form, the expression has two terms that depend on \( x \): \( 2x^2 \) and \( -5x \).
  • It is not immediately clear how the value of \( x \) influences the entire expression.

Completed Square Form: \( 2\left(x - \frac{5}{4}\right)^2 - \frac{33}{8} \)

  • After completing the square, the expression becomes \( 2\left(x - \frac{5}{4}\right)^2 - \frac{33}{8} \).
  • Here, there is only one term that depends on \( x \), namely \( \left(x - \frac{5}{4}\right)^2 \).
  • This form clearly shows that the expression's value is influenced primarily by how \( x \) deviates from \( \frac{5}{4} \). It's easier to see the effect of changing \( x \) on the whole expression.

Conclusion

  • The transformation into the completed square form simplifies the expression, reducing it to a single \( x \)-dependent term, which makes it much easier to analyze and understand.
  • It provides how changes in \( x \) affect the function's value. (if not, then let us wait for the upcoming sections)

Thus, while the original form is useful for certain types of analysis and solving, the completed square form offers a more direct insight into the function's graphical characteristics and behavior.

Practice Problems

Convert the Following Quadratic Expressions into Completed Square Form

Convert each of the following quadratic expressions into its completed square form:

  1. \( x^2 + 6x + 5 \)
  2. \( 3x^2 - 12x + 7 \)
  3. \( -2x^2 + 8x - 5 \)
  4. \( 4x^2 - 16x + 15 \)

Answers

  1. \( x^2 + 6x + 5 \): Completed Square Form: \( (x + 3)^2 - 4 \)

  2. \( 3x^2 - 12x + 7 \): Completed Square Form: \( 3(x - 2)^2 - 5 \)

  3. \( -2x^2 + 8x - 5 \): Completed Square Form: \( -2(x - 2)^2 + 3 \)

  4. \( 4x^2 - 16x + 15 \): Completed Square Form: \( 4(x - 2)^2 + 1 \)