Problem 32 Use completing the square to wri... [FREE SOLUTION] (2024)

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Quadratic equations are polynomial equations of degree 2. This means the highest exponent of the variable (usually denoted as x) is 2. The standard form is \( ax^2 + bx + c = 0 \).

In this equation:

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

The solutions to a quadratic equation can be found using various methods like:

• Factoring
• Completing the square
• Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In cases where we need to analyze parabolas, it's often useful to express quadratic equations in the vertex form, which reveals more about the graph's properties.

Converting a quadratic equation to its vertex form helps us easily find key features of a parabola, such as its vertex. The vertex form of a quadratic equation is \( y = a(x-h)^2 + k \).

In this form:

• \((h, k)\) is the vertex of the parabola.
• \(a\) determines the direction and width of the parabola:
• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

To convert a standard form quadratic to vertex form, we use the method of completing the square. Let's take an example:

• Given: \(y = x^2 + 2x - 5\)
• Rewriting it by completing the square:
• \(y = (x + 1)^2 - 6\)

This vertex form shows that the parabola has a vertex at \((-1, -6)\).

A parabola can also be described in terms of its focus and directrix.

The focus of a parabola is a point from which distances to points on the parabola are measured. The directrix is a line such that the distance of any point on the parabola to the directrix is equal to its distance to the focus. For a parabola in the vertex form \( y = a(x-h)^2 + k \):

• \( h \) and \( k \) give us the vertex.
• The distance \( p \) from the vertex to the focus is found using \( 4p = 1/a \).

In our example \( y = (x + 1)^2 - 6 \), \( a \) is 1 which means:

• \( 4p = 1 \), so \( p = 1/4 \).
• Focus is at \( ( -1, -6 + \frac{1}{4}) = (-1, -5.75) \).
• Directrix is at \( y = -6 - \frac{1}{4} = -6.25 \).

Understanding how these elements work together gives us deeper insight into the behavior and properties of parabolas.