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Completing the Square quiz

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  • What is the preferred form of a quadratic equation before completing the square?
    The preferred form is x^2 + bx = c, where the leading coefficient is 1 and the constant is isolated on the right side.
  • What value do you add to both sides when completing the square?
    You add (b/2)^2 to both sides, where b is the coefficient of x.
  • Why do we add (b/2)^2 to both sides of the equation?
    Adding (b/2)^2 creates a perfect square trinomial, which can be factored into a binomial squared.
  • How do you factor x^2 + 6x + 9 after completing the square?
    It factors into (x + 3)^2, since 6/2 is 3 and 3^2 is 9.
  • What is the next step after factoring the perfect square trinomial?
    The next step is to use the square root property to solve for x.
  • What does the square root property involve when solving (x + 3)^2 = 2?
    It involves taking the positive and negative square roots of both sides, resulting in x + 3 = ±√2.
  • How do you isolate x after applying the square root property?
    Subtract the constant from both sides, so x = -3 ± √2.
  • What is the solution to x^2 + 6x = -7 after completing the square?
    The solution is x = -3 ± √2.
  • When is completing the square especially useful?
    It is especially useful when the leading coefficient is 1 and the x-term coefficient is even.
  • What is a perfect square trinomial?
    A perfect square trinomial is an expression that can be factored into (x + a)^2 for some value a.
  • What is the first step in completing the square for x^2 + 8x + 1 = 0?
    Move the constant to the other side to get x^2 + 8x = -1.
  • What value should be added to both sides for x^2 + 8x = -1?
    Add (8/2)^2 = 16 to both sides.
  • How does x^2 + 8x + 16 factor after completing the square?
    It factors into (x + 4)^2.
  • What is the equation after completing the square for x^2 + 8x + 1 = 0?
    The equation becomes (x + 4)^2 = 15.
  • What is the general process for completing the square?
    Rewrite the equation in x^2 + bx = c form, add (b/2)^2 to both sides, factor, and solve using the square root property.