The two formulas you've mentioned are related to quadratic equations, but they serve different purposes and are not the same. Let's clarify their uses:

1. Quadratic Formula for Solving $x$:

The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ is used to find the roots (solutions) of a quadratic equation of the form $ax^2 + bx + c = 0$. This formula gives you the $x$-values where the parabola intersects the $x$-axis.

2. Formula for Finding the Vertex of a Parabola:

The formula $x = \frac{-b}{2a}$ is used to find the $x$-coordinate of the vertex of the parabola represented by the quadratic equation $y = ax^2 + bx + c$. Once you find this $x$-coordinate, you can substitute it back into the original equation to find the $y$-coordinate of the vertex.

When to Use Each Formula:

• Quadratic Formula (1):

  • Use this when you need to find the roots of the quadratic equation.
  • Example: You want to know where the graph intersects the $x$-axis.

• Vertex Formula (2):

  • Use this when you need to find the vertex of the parabola.
  • Example: You want to know the highest or lowest point on the graph, which is particularly useful in optimization problems.

Example for Clarity:

Consider the quadratic equation $y = 2x^2 - 4x + 1$.

Finding the Roots:

Using the quadratic formula: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$ $x = \frac{4 \pm \sqrt{16 - 8}}{4}$ $x = \frac{4 \pm \sqrt{8}}{4}$ $x = \frac{4 \pm 2\sqrt{2}}{4}$ $x = 1 \pm \frac{\sqrt{2}}{2}$

So, the roots are $x = 1 + \frac{\sqrt{2}}{2}$ and $x = 1 - \frac{\sqrt{2}}{2}$.

Finding the Vertex:

Using the vertex formula: $x = \frac{-(-4)}{2 \cdot 2}$ $x = \frac{4}{4}$ $x = 1$

Now, substitute $x = 1$ back into the original equation to find the $y$-coordinate of the vertex: $y = 2(1)^2 - 4(1) + 1$ $y = 2 - 4 + 1$ $y = -1$

So, the vertex of the parabola is at $(1, -1)$.

Summary:

• Use the quadratic formula to find the roots of the quadratic equation.
• Use the vertex formula to find the vertex of the parabola.

Would you like more details on any of these points or have any questions?

Questions for You to Consider:

1. Do you want to see more examples of using these formulas?
2. Do you need help with a specific quadratic equation?
3. Are you interested in learning how these formulas are derived?
4. Would you like to understand the graphical interpretation of these formulas?
5. Do you need assistance with solving a different type of mathematical problem?

Tip:

When solving quadratic equations, always check if the discriminant ($b^2 - 4ac$) is positive, zero, or negative, as it tells you the nature of the roots (real and distinct, real and repeated, or complex).