Quadratic equations examples

📚 Quick Study Guide

• 🔍 Standard Form: A quadratic equation is typically written as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \neq 0$.
• ➗ Solving by Factoring: Factor the quadratic expression into two binomials, then set each binomial equal to zero and solve for $x$. For example, $x^2 + 5x + 6 = (x+2)(x+3)=0$, so $x=-2$ or $x=-3$.
• 💡 Quadratic Formula: When factoring is difficult or impossible, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• 📈 Discriminant: The discriminant, $b^2 - 4ac$, determines the nature of the roots:
  • If $b^2 - 4ac > 0$, there are two distinct real roots.
  • If $b^2 - 4ac = 0$, there is one real root (a repeated root).
  • If $b^2 - 4ac < 0$, there are two complex roots.
• 📝 Completing the Square: A method to rewrite the quadratic equation into the form $(x + p)^2 = q$, making it easier to solve for $x$.

🧪 Practice Quiz

1. What are the solutions to the quadratic equation $x^2 - 5x + 6 = 0$?
   1. x = 2, x = 3
   2. x = -2, x = -3
   3. x = 1, x = 6
   4. x = -1, x = -6
2. Solve for $x$: $2x^2 + 7x + 3 = 0$
   1. x = -3, x = -1/2
   2. x = 3, x = 1/2
   3. x = -3, x = 1/2
   4. x = 3, x = -1/2
3. What is the discriminant of the quadratic equation $3x^2 - 4x + 2 = 0$?
   1. -8
   2. 8
   3. 40
   4. -40
4. How many real solutions does the equation $x^2 + 6x + 9 = 0$ have?
   1. 0
   2. 1
   3. 2
   4. 3
5. Find the roots of the equation $x^2 - 9 = 0$.
   1. x = 3, x = -3
   2. x = 9, x = -9
   3. x = 3
   4. x = -3
6. Solve $x^2 + 2x - 8 = 0$ by factoring.
   1. x = 2, x = -4
   2. x = -2, x = 4
   3. x = 1, x = -8
   4. x = -1, x = 8
7. Using the quadratic formula, find the solutions to $x^2 + 4x + 1 = 0$.
   1. x = -2 ± \sqrt{3}
   2. x = 2 ± \sqrt{3}
   3. x = -4 ± \sqrt{5}
   4. x = 4 ± \sqrt{5}