Step-by-step examples for writing quadratic equations in Algebra 2

📚 Quick Study Guide

🔍 A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. 💡 To write a quadratic equation given roots $r_1$ and $r_2$, use the form $a(x - r_1)(x - r_2) = 0$. 📝 If given the sum ($S$) and product ($P$) of the roots, the equation can be written as $x^2 - Sx + P = 0$. ➗ Completing the square involves manipulating the equation to form a perfect square trinomial. ➕ The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to find the roots of any quadratic equation.

🧪 Practice Quiz

1. What quadratic equation has roots 3 and -2?
   1. $x^2 - x - 6 = 0$
   2. $x^2 + x - 6 = 0$
   3. $x^2 - 5x + 6 = 0$
   4. $x^2 + 5x + 6 = 0$
2. If the sum of the roots of a quadratic equation is 5 and the product is 6, what is the equation?
   1. $x^2 + 5x + 6 = 0$
   2. $x^2 - 5x + 6 = 0$
   3. $x^2 + 5x - 6 = 0$
   4. $x^2 - 5x - 6 = 0$
3. Which quadratic equation has a single real root?
   1. $x^2 + 4x + 3 = 0$
   2. $x^2 + 4x + 4 = 0$
   3. $x^2 + 4x + 5 = 0$
   4. $x^2 + 4x = 0$
4. What is the quadratic equation that passes through the point (0, -8) and has roots 2 and -4?
   1. $x^2 + 2x - 8 = 0$
   2. $x^2 - 2x - 8 = 0$
   3. $2x^2 + 4x - 16 = 0$
   4. $x^2 + 4x - 8 = 0$
5. Write a quadratic equation where one root is 2 + i.
   1. $x^2 - 4x + 5 = 0$
   2. $x^2 + 4x + 5 = 0$
   3. $x^2 - 4x - 5 = 0$
   4. $x^2 + 4x - 5 = 0$
6. What quadratic equation has roots that are reciprocals of each other and sum to $\frac{5}{2}$?
   1. $2x^2 - 5x + 2 = 0$
   2. $2x^2 + 5x + 2 = 0$
   3. $x^2 - 5x + 1 = 0$
   4. $x^2 + 5x + 1 = 0$
7. A rectangle has a length that is 3 more than its width. If the area is 10, what quadratic equation can be used to find the width, $w$?
   1. $w^2 + 3w - 10 = 0$
   2. $w^2 - 3w - 10 = 0$
   3. $w^2 + 3w + 10 = 0$
   4. $w^2 - 3w + 10 = 0$