🔹 Solving Quadratic Equations
Master multiple methods to solve quadratic equations! From factoring to the powerful quadratic formula.
🎯 What is a Quadratic Equation?
A quadratic equation has the form $ax^2 + bx + c = 0$ where $a \neq 0$.
Examples:
- $x^2 - 5x + 6 = 0$
- $2x^2 + 3x - 5 = 0$
- $x^2 = 9$ (can be written as $x^2 - 9 = 0$)
⚡ Solving by Factoring
If you can factor the quadratic, set each factor equal to zero!
Example: $x^2 + 5x + 6 = 0$
Step 1: Factor: $(x + 2)(x + 3) = 0$
Step 2: Set each factor to zero:
$x + 2 = 0$ or $x + 3 = 0$
Solutions: $x = -2$ or $x = -3$ ✨
√ Square Root Method
For equations like $x^2 = k$, use square roots!
Example: $x^2 = 25$
Take the square root of both sides:
$$x = \pm 5$$
Solutions: $x = 5$ or $x = -5$
✨ Completing the Square
Transform the equation into $(x + p)^2 = q$ form!
Example: $x^2 + 6x + 5 = 0$
Step 1: Move constant: $x^2 + 6x = -5$
Step 2: Add $(\frac{6}{2})^2 = 9$ to both sides:
$x^2 + 6x + 9 = 4$
Step 3: Factor: $(x + 3)^2 = 4$
Step 4: Solve: $x = -3 \pm 2$ → $x = -1$ or $x = -5$
🔮 The Quadratic Formula
The universal solution! Works for ANY quadratic equation!
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Example: $2x^2 + 5x - 3 = 0$
Here $a=2$, $b=5$, $c=-3$
$$x = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm 7}{4}$$
Solutions: $x = \frac{1}{2}$ or $x = -3$
🎯 Practice Questions
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🔥 Challenge Questions
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