Definition
A quadratic equation is any equation of the form
$$
ax^2 + bx + c = 0
$$
where $a$, $b$, and $c$ are constants, and $a \neq 0$.
The solutions (roots) of a quadratic equation can be found using the quadratic formula:
$$
x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}
$$
Worked Example
Solve: $2x^2 - 4x - 6 = 0$ **Step 1:** Identify $a$, $b$, and $c$: - $a = 2$ - $b = -4$ - $c = -6$ **Step 2:** Plug into the quadratic formula: %%MATH_DISPLAY_2%% **Step 3:** Simplify inside the square root: %%MATH_DISPLAY_3%% **Step 4:** Substitute back: %%MATH_DISPLAY_4%% **Step 5:** Calculate the roots: - $sqrt{64} = 8$ So, %%MATH_DISPLAY_5%% **Final Answer:** $x = 3$ and $x = -1$ ## Takeaways - Quadratic equations have at most two real solutions. - Use the quadratic formula for any quadratic equation: $x = frac{ -b pm sqrt{b^2 - 4ac} }{2a}$. - The discriminant ($b^2 - 4ac$) determines the nature of the roots (real or complex).