Completing the square

• What is the process of rewriting the first two terms of a quadratic expression as the difference of two squares called?

  Completing the Square

  
• How can the quadratic expression \(x^2 + bx\) be rewritten?

  As \((x + p)^2 - p^2\) where \(p\) is half of \(b\)

  
• If \(b = 2\), how would you express \(x^2 + 2x\) using the difference of two squares?


  \((x + 1)^2 - 1\)

  
• What is the result of expanding \((x + 1)^2 - 1\)?

  It simplifies to \(x^2 + 2x\)

  
• How can the expression \(x^2 - 20x\) be rewritten using the difference of two squares?

  As \((x - 10)^2 - 100\)

  
• What is the first step in completing the square for the expression \(x^2 + 10x + 9\)?

  Replace \(x^2 + 10x\) with \((x + 5)^2 - 25\)

  
• What is the final expression after completing the square for \(x^2 + 10x + 9\)?


  \((x + 5)^2 - 16\)

  
• What should you do first when completing the square for an expression with a coefficient in front of the \(x\) term?


  Factor out \(a\) from the \(x^2\) and \(x\) terms

  
• How is the expression \(ax^2 + bx + c\) rewritten when completing the square?

  As \(a\left(x^2 + \frac{b}{a}x\right) + c\)

  
• What is the form of the expression after completing the square inside the square brackets?

  \(a\left(x + p\right)^2 - ap^2 + c\)

  
• What is the final form of the expression after completing the square?

  In the form \(a\left(x + p\right)^2 + q\)

  
• How can completing the square help find the turning point of a quadratic graph?


  It allows us to express the quadratic in the form \(y = (x + p)^2 + q\)

  
• What are the coordinates of the turning point when \(y = (x + p)^2 + q\)?

  \((-p, q)\)

  
• What does the sign of \(a\) indicate about the turning point of the quadratic function?


  If \(a > 0\), it is a minimum point; if \(a < 0\), it is a maximum point

  
• How can completing the square be used to create the equation of a quadratic when given the turning point?


  By using the turning point coordinates to form \(y = a(x + p)^2 + q\)

  
• Why is it important to check your completed square by expanding your answer?


  To ensure that you have completed the square correctly
• What is the first step to find the turning point of the graph \(y = x^2 + 6x - 11\)?

  Find half of \(6\) (call this \(p\))

  
• What is the value of \(p\) when finding the turning point of \(y = x^2 + 6x - 11\)?

  3

  
• How do you express \(x + 6x\) in the form \((x + p)^2 - p\)?


  As \((x + 3)^2 - 9\)

  
• What is the turning point of the quadratic \(y = x^2 + 6x - 11\) after completing the square?

  The turning point is at \((-3, -20)\)

  
• How do you rewrite \(-3x^2 + 12x + 24\) in the form \(a(x + p)^2 + q\)?

  Factor out \(-3\) from the first two terms only

  
• What is the form of the expression after completing the square for \(-3x^2 + 12x + 24\)?

  In the form \(-3(x + p)^2 + q\)

  
• What is the value of \(p\) when completing the square for \(-3x^2 + 12x + 24\)?


  • 2

  
• What is the final value of \(q\) after completing the square for \(-3x^2 + 12x + 24\)?

  36

  
• What are the steps to complete the square for a quadratic expression?

  1. Rewrite the first two terms as the difference of two squares.
  2. Factor out any coefficient from the \(x^2\) and \(x\) terms.
  3. Complete the square inside the brackets.
  4. Multiply by the coefficient and add the constant term.
  5. Write the final expression in the form \(a(x + p)^2 + q\).

  
• What is the significance of the turning point in a quadratic graph?

  • Indicates the maximum or minimum value of the quadratic function.
  • Helps in graphing the quadratic function.
  • Provides insight into the behavior of the function.

  
• What is the relationship between the coefficients and the turning point of a quadratic function?

  • The turning point is at \((-p, q)\) where \(p\) is half of the coefficient of \(x\).
  • The sign of \(a\) determines if the turning point is a maximum or minimum.