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Completing the Square quiz

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  • What is the main goal of completing the square when solving a quadratic equation?
    The goal is to rewrite the equation in the form (x + a number)^2 = constant so the square root property can be used to solve for x.
  • When is completing the square especially useful for solving quadratic equations?
    It is especially useful when the leading coefficient (a) is 1 and the coefficient of x (b) is even.
  • What is the first step in completing the square for a quadratic equation?
    Rearrange the equation into the form x^2 + bx = c, isolating the constant on one side.
  • What value do you add to both sides of the equation to complete the square?
    Add (b/2)^2 to both sides, where b is the coefficient of x.
  • Why do we add (b/2)^2 to both sides when completing the square?
    Adding (b/2)^2 creates a perfect square trinomial on one side, which can be factored into (x + b/2)^2.
  • After adding (b/2)^2 to both sides, how do you rewrite the left side of the equation?
    Factor the left side as (x + b/2)^2.
  • What property do you use to solve for x after completing the square?
    Use the square root property, taking both the positive and negative square roots of both sides.
  • In the example x^2 + 6x = -7, what value is added to both sides to complete the square?
    9 is added to both sides, since (6/2)^2 = 9.
  • What is the factored form of x^2 + 6x + 9?
    It factors to (x + 3)^2.
  • After completing the square for x^2 + 6x = -7, what equation do you get before solving for x?
    You get (x + 3)^2 = 2.
  • How do you solve (x + 3)^2 = 2 for x?
    Take the square root of both sides to get x + 3 = ±√2, then subtract 3 to isolate x.
  • What are the solutions to x^2 + 6x = -7 after completing the square?
    The solutions are x = -3 ± √2.
  • In the example x^2 + 8x + 1 = 0, what is the first step to complete the square?
    Move the constant to the other side to get x^2 + 8x = -1.
  • What value is added to both sides when completing the square for x^2 + 8x = -1?
    16 is added to both sides, since (8/2)^2 = 16.
  • What is the factored form and resulting equation after completing the square for x^2 + 8x = -1?
    The equation becomes (x + 4)^2 = 15.