Problem 46 - Algebra 2

Question

Write in the form \(y=a(x-h)^{2}+k\) $$ y=x^{2}+6 x+9 $$

Step-by-Step Solution

Answer

The vertex form is \(y = (x+3)^2\).

Step 1: Identify the quadratic form components

The given quadratic is in the standard form: \(y = ax^2 + bx + c\), where \(a = 1\), \(b = 6\), and \(c = 9\). We need to convert it into the vertex form \(y = a(x-h)^2+k\).

Step 2: Complete the square

To complete the square, we focus on the terms \(x^2 + 6x\). We need to add and subtract \((b/2)^2 = (6/2)^2 = 9\) inside the quadratic expression to form a perfect square trinomial. Hence, \(x^2 + 6x + 9\) becomes \((x + 3)^2\).

Step 3: Rewrite in vertex form

After completing the square, we rewrite the expression as \(y = (x + 3)^2\). Since there is no additional constant term outside the square, \(k = 0\), and \(a = 1\). Thus, the expression in vertex form is \(y = 1(x + 3)^2 + 0\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two. This typically means it will have the general form: \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). The presence of the squared term \( x^2 \) gives the quadratic its characteristic parabolic shape when graphed.
Quadratics can be used to model a variety of real-world situations, from calculating projectile motion to determining areas. In our exercise, we deal with a specific quadratic equation \( y = x^2 + 6x + 9 \), which we'll later convert to vertex form.

Completing the Square

Completing the square is a method used to convert a quadratic equation from its standard form to vertex form. This technique can unveil a quadratic's maximum or minimum value, depending on the direction of its parabola.

Here’s how it works:

• First, identify \( b \), the coefficient of the \( x \) term in the equation \( ax^2 + bx + c \).
• Take half of \( b \), square it, and then add and subtract it within the equation. This forms a perfect square trinomial.

For our example with \( y = x^2 + 6x + 9 \), we realized that \( 6 / 2 = 3 \) and \( 3^2 = 9 \). By adding and subtracting 9, we reformulate \( x^2 + 6x + 9 \) as \( (x + 3)^2 \), demonstrating how completing the square helps reshape our equation.

Standard Form

The standard form of a quadratic equation is \( y = ax^2 + bx + c \). This format is helpful for quickly identifying the coefficients that dictate the behavior of the parabola, such as its width and direction.

By observing the given quadratic equation, \( y = x^2 + 6x + 9 \), we see the direct alignment with standard form where \( a = 1 \), \( b = 6 \), and \( c = 9 \). Standard form is the starting point before transforming it into vertex form or other representations.
Understanding standard form is crucial as it serves as the segue to converting equations through methods like completing the square.

Perfect Square Trinomial

A perfect square trinomial is a unique type of expression derived from squaring a binomial. Essentially, it takes the form \((x + d)^2 \), and upon expansion, becomes \(x^2 + 2dx + d^2\). These trinomials are perfect for rewriting quadratic expressions in vertex form.

In our problem, the expression \( x^2 + 6x + 9 \) was identified as a perfect square trinomial because it can be rewritten as \((x + 3)^2\). The term \(x^2 + 6x + 9\) follows the expansion of \((x + 3)\), ensuring the equivalency necessary for vertex form.
Recognizing and using perfect square trinomials simplifies the process of handling and analyzing quadratics.