Definition
A quadratic equation is any equation that can be written in the form:
$$
ax^2 + bx + c = 0
$$
where $a$, $b$, and $c$ are constants, and $a \neq 0$.
The general method to solve a quadratic equation is to use 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:** Substitute into the quadratic formula: %%MATH_DISPLAY_2%% **Step 3:** Simplify inside the square root: %%MATH_DISPLAY_3%% **Step 4:** Add inside the square root: %%MATH_DISPLAY_4%% **Step 5:** Take the square root: %%MATH_DISPLAY_5%% **Step 6:** Solve for both $+$ and $-$ cases: - $x_1 = frac{4 + 8}{4} = frac{12}{4} = 3$ - $x_2 = frac{4 - 8}{4} = frac{-4}{4} = -1$ **Final Answer:** $x = 3$ or $x = -1$ --- ## Takeaways - Quadratic equations can be solved using the quadratic formula. - Always identify $a$, $b$, and $c$ before substituting. - The discriminant $b^2 - 4ac$ determines the nature of the solutions.