Examples of solving quadratics by completing the square step-by-step

📚 Quick Study Guide

• 🔢 The goal of completing the square is to rewrite a quadratic equation in the form $ax^2 + bx + c = 0$ into the form $a(x + h)^2 + k = 0$.
• ➕ To complete the square, take half of the coefficient of the $x$ term (which is $b$), square it, and add it to both sides of the equation. Specifically, add $(\frac{b}{2})^2$.
• ✏️ Rewrite the quadratic as a perfect square trinomial. This will be in the form $(x + \frac{b}{2})^2$.
• ⚖️ Solve for $x$ by taking the square root of both sides and isolating $x$. Remember to consider both positive and negative square roots.
• 💡 If $a \neq 1$, divide the entire equation by $a$ before completing the square.

Practice Quiz

1. What value should be added to $x^2 + 6x$ to complete the square?
   1. 3
   2. 6
   3. 9
   4. 12
2. Solve for $x$ by completing the square: $x^2 + 4x - 5 = 0$
   1. $x = 1, -5$
   2. $x = -1, 5$
   3. $x = -1, -5$
   4. $x = 1, 5$
3. What is the vertex form of the quadratic equation $y = x^2 - 8x + 15$ after completing the square?
   1. $y = (x - 4)^2 - 1$
   2. $y = (x + 4)^2 - 1$
   3. $y = (x - 4)^2 + 1$
   4. $y = (x + 4)^2 + 1$
4. Solve for $x$ by completing the square: $2x^2 - 8x + 6 = 0$
   1. $x = 1, 3$
   2. $x = -1, -3$
   3. $x = 1, -3$
   4. $x = -1, 3$
5. Which of the following is the correct first step in completing the square for the equation $3x^2 + 12x - 15 = 0$?
   1. Divide the entire equation by 3.
   2. Add 15 to both sides.
   3. Divide the entire equation by 12.
   4. Subtract 12 from both sides.
6. Find the value of $c$ that makes $x^2 - 5x + c$ a perfect square trinomial.
   1. $\frac{25}{4}$
   2. $\frac{5}{2}$
   3. 25
   4. 10
7. Solve for $x$ by completing the square: $x^2 + 2x = 8$
   1. $x = 2, -4$
   2. $x = -2, 4$
   3. $x = 2, 4$
   4. $x = -2, -4$