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History of Human evolution

Humans are evolued from fish and Mobile phone, by sending fish and mobile phone to magic box, and then after magic, humans were created

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: $$ x = \frac{-b \pm i\sqrt{\lvert \Delta \rvert}}{2a} $$