Algebra 1: Quadratic Formula - Examples and Problems

Understanding the Quadratic Formula 🧮

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \ne 0$. The formula is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Steps to Use the Quadratic Formula

1. Identify a, b, and c: Rewrite the equation in the standard form $ax^2 + bx + c = 0$ and identify the coefficients $a$, $b$, and $c$.
2. Substitute: Plug the values of $a$, $b$, and $c$ into the quadratic formula.
3. Simplify: Simplify the expression, paying close attention to the $\pm$ sign, which indicates two possible solutions.
4. Solve for x: Calculate the two possible values of $x$.

Example Problems with Solutions 🚀

Example 1: Solve $x^2 - 5x + 6 = 0$

1. Identify: $a = 1$, $b = -5$, $c = 6$
2. Substitute: $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}$
3. Simplify: $x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$
4. Solve: $x_1 = \frac{5 + 1}{2} = 3$, $x_2 = \frac{5 - 1}{2} = 2$

Therefore, the solutions are $x = 3$ and $x = 2$.

Example 2: Solve $2x^2 + 4x - 6 = 0$

1. Identify: $a = 2$, $b = 4$, $c = -6$
2. Substitute: $x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-6)}}{2(2)}$
3. Simplify: $x = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm \sqrt{64}}{4} = \frac{-4 \pm 8}{4}$
4. Solve: $x_1 = \frac{-4 + 8}{4} = 1$, $x_2 = \frac{-4 - 8}{4} = -3$

Therefore, the solutions are $x = 1$ and $x = -3$.

Practice Problems ✍️

• Solve $x^2 + 2x - 8 = 0$
• Solve $3x^2 - 6x - 9 = 0$
• Solve $x^2 - 4x + 4 = 0$

Solutions to Practice Problems ✅

1. For $x^2 + 2x - 8 = 0$: $a = 1, b = 2, c = -8$. $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-8)}}{2(1)} = \frac{-2 \pm \sqrt{36}}{2} = \frac{-2 \pm 6}{2}$. $x = 2, -4$
2. For $3x^2 - 6x - 9 = 0$: $a = 3, b = -6, c = -9$. $x = \frac{6 \pm \sqrt{(-6)^2 - 4(3)(-9)}}{2(3)} = \frac{6 \pm \sqrt{144}}{6} = \frac{6 \pm 12}{6}$. $x = 3, -1$
3. For $x^2 - 4x + 4 = 0$: $a = 1, b = -4, c = 4$. $x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(4)}}{2(1)} = \frac{4 \pm \sqrt{0}}{2} = 2$. $x = 2$ (repeated root)