Test questions on Solving Quadratics by Completing the Square

📚 Quick Study Guide

• 🔢 A quadratic equation is in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.
• ✍️ Completing the square involves manipulating the quadratic equation to create a perfect square trinomial on one side.
• ➕ To complete the square for $x^2 + bx$, add $(\frac{b}{2})^2$ to both sides of the equation.
• 📝 The completed square form is $(x + \frac{b}{2})^2$.
• ✅ Steps:
  1. If $a \neq 1$, divide the entire equation by $a$.
  2. Move the constant term to the right side of the equation.
  3. Add $(\frac{b}{2})^2$ to both sides.
  4. Factor the left side as a perfect square.
  5. Take the square root of both sides.
  6. Solve for $x$.

🧪 Practice Quiz

1. What value of $c$ completes the square for $x^2 + 6x + c$?
   1. 3
   2. 6
   3. 9
   4. 12
2. Solve for $x$ by completing the square: $x^2 + 4x - 5 = 0$
   1. $x = -5, 1$
   2. $x = 5, -1$
   3. $x = -5, -1$
   4. $x = 5, 1$
3. What should be added to both sides to complete the square for $x^2 - 8x = 2$?
   1. 4
   2. 8
   3. 16
   4. 64
4. Rewrite $x^2 - 2x + 5 = 0$ by completing the square.
   1. $(x - 1)^2 = -4$
   2. $(x + 1)^2 = -4$
   3. $(x - 1)^2 = 4$
   4. $(x + 1)^2 = 4$
5. Solve $2x^2 + 8x - 10 = 0$ by completing the square.
   1. $x = -5, 1$
   2. $x = 5, -1$
   3. $x = -1, 5$
   4. $x = 1, -5$
6. Which equation is equivalent to $(x - 3)^2 = 7$ after completing the square?
   1. $x^2 - 6x + 2 = 0$
   2. $x^2 - 6x - 2 = 0$
   3. $x^2 + 6x + 2 = 0$
   4. $x^2 + 6x - 2 = 0$
7. Find the vertex form of the quadratic $y = x^2 + 10x + 20$ by completing the square.
   1. $y = (x + 5)^2 - 5$
   2. $y = (x - 5)^2 - 5$
   3. $y = (x + 5)^2 + 5$
   4. $y = (x - 5)^2 + 5$