Quadratic Formula: Definition
The quadratic formula is a method for solving quadratic equations of the form
$$ ax^2 + bx + c = 0 $$
where $a$, $b$, and $c$ are constants, and $a \neq 0$. The solutions for $x$ are given by:
$$ x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} $$
Here, the symbol $\pm$ means there are generally two solutions: one with $+$ and one with $-$.
Worked Example
Solve: $2x^2 - 4x - 6 = 0$ **Step 1: Identify coefficients** - $a = 2$ - $b = -4$ - $c = -6$ **Step 2: Substitute into the quadratic formula** %%MATH_DISPLAY_2%% **Step 3: Simplify inside the square root** - $(-4)^2 = 16$ - $4 cdot 2 cdot (-6) = -48$ So, %%MATH_DISPLAY_3%% %%MATH_DISPLAY_4%% %%MATH_DISPLAY_5%% **Step 4: Simplify the square root** - $sqrt{64} = 8$ So, %%MATH_DISPLAY_6%% **Step 5: Find both solutions** - $x_1 = frac{4 + 8}{4} = frac{12}{4} = 3$ - $x_2 = frac{4 - 8}{4} = frac{-4}{4} = -1$ **Final Answer:** $x = 3$ or $x = -1$ --- ## Takeaways - The quadratic formula solves any quadratic equation $ax^2 + bx + c = 0$. - Always substitute $a$, $b$, and $c$ carefully and simplify step by step. - The discriminant $b^2 - 4ac$ determines the nature of the solutions (real or complex).