91Ó°ÊÓ

The vertex form of a quadratic equation is a useful way to express a quadratic function, especially when you know the vertex. The vertex form is written as:
\[ f(x) = a(x - h)^2 + k \]
Here, \((h, k)\) represents the vertex of the parabola.

For example, in the exercise above, the vertex \((5, 3)\) was provided. Plugging these values into the vertex form, we get:
\[ f(x) = a(x - 5)^2 + 3 \]
The constant ‘a’ determines the parabola's direction and width. We found ‘a’ by substituting a known solution (\( x = 2 \)) and rearranging to solve for ‘a’. This gave us:
\[ 0 = a(2 - 5)^2 + 3 \Rightarrow a = -\frac{1}{3} \]
So, the equation becomes:
\[ f(x) = -\frac{1}{3}(x - 5)^2 + 3 \]
Understanding the vertex form is key to easily recognize the vertex and make calculations with known points on the graph.

Converting the vertex form of a quadratic equation to the standard form can be useful for solving the equation or understanding its properties.
The standard form of a quadratic equation is given by:
\[ ax^2 + bx + c \]
Let's use the vertex form from the exercise:
\[ f(x) = -\frac{1}{3}(x - 5)^2 + 3 \]
First, expand the squared term:
\[ (x - 5)^2 = x^2 - 10x + 25 \]
Now substitute the expanded term back into the equation:
\[ f(x) = -\frac{1}{3}(x^2 - 10x + 25) + 3 \]
Distribute -\frac{1}{3} to each term in the parenthesis:
\[ f(x) = -\frac{1}{3}x^2 + \frac{10}{3}x - \frac{25}{3} + 3 \]
Simplify the constant terms:
\[ 3 = \frac{9}{3} \]
So, the equation simplifies to:
\[ f(x) = -\frac{1}{3}x^2 + \frac{10}{3}x - \frac{16}{3} \]
Converting between forms allows you to access different properties of the quadratic function, such as its roots or the y-intercept.

Factoring a quadratic equation is an essential skill for finding its roots. Here's a step-by-step guide using the standard form derived previously:
\[ f(x) = -\frac{1}{3}x^2 + \frac{10}{3}x - \frac{16}{3} = 0 \]
First, clear the fraction by multiplying every term by -3:
\[ x^2 - 10x + 16 = 0 \]
Now, factor the quadratic equation. We look for two numbers that multiply to +16 and add up to -10. These numbers are -2 and -8:
\[ (x - 2)(x - 8) = 0 \]
Set each factor equal to zero to find the solutions:
\[ x - 2 = 0 \Rightarrow x = 2 \]
\[ x - 8 = 0 \Rightarrow x = 8 \]
Therefore, the solutions are \(x = 2\thinspace\text{and}\thinspace x = 8 \). Factoring shows how changes in the equation's coefficients affect its roots.
Understanding factoring helps you break down and solve more complex quadratic equations efficiently.