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# Solving Quadratic Equations

Quadratic equations are equations of the form: $$ ax^2 + bx + c = 0 $$ where $a$, $b$, and $c$ are constants, and $a \neq 0$.

General Solution

The general solution to a quadratic equation is given by the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Steps to Solve

1. Identify coefficients: From the quadratic equation $ax^2 + bx + c = 0$, determine $a$, $b$, and $c$.

2. Compute the discriminant: $$ \Delta = b^2 - 4ac $$

   • If $\Delta > 0$: Two distinct real solutions.
   • If $\Delta = 0$: One real solution (a repeated root).
   • If $\Delta < 0$: Two complex solutions.

3. Apply the quadratic formula: Substitute $a$, $b$, and $\Delta$ into the formula to find the solutions for $x$.

Example

Solve the quadratic equation: $$ 2x^2 - 4x - 6 = 0 $$

Step 1: Identify coefficients

$a = 2$, $b = -4$, $c = -6$.

Step 2: Compute the discriminant

$$ \Delta = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 $$

Step 3: Apply the quadratic formula

$$ x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4} $$

Solutions:

$$ x_1 = \frac{4 + 8}{4} = 3, \quad x_2 = \frac{4 - 8}{4} = -1 $$

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

Special Cases

1. Perfect Square Trinomial: If $\Delta = 0$, the equation has one real root: $$ x = \frac{-b}{2a} $$

2. Complex Roots: If $\Delta < 0$, use $i$ to express the solutions: $$ x = \frac{-b \pm i\sqrt{\lvert \Delta \rvert}}{2a} $$