Problem 66 - Algebra 2

Question

Write a quadratic equation with the given roots. Write the equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are integers. \(6,-6\)

Step-by-Step Solution

Answer

The quadratic equation is \( x^2 - 36 = 0 \).

Step 1: Understand the Roots

We are given the roots of the quadratic equation as 6 and -6. Recall that if \( p \) and \( q \) are the roots of a quadratic equation, then the equation can be written as \((x - p)(x - q) = 0\). In this case, the roots are 6 and -6, which means our equation becomes \((x - 6)(x + 6) = 0\).

Step 2: Express in standard form

The quadratic equation obtained from the roots must be expanded to express it in the standard form \( ax^2 + bx + c = 0 \). Calculate the product: \((x - 6)(x + 6)\). This expression is an example of the difference of squares formula, \( a^2 - b^2 = (a+b)(a-b) \). Here, \( a = x \) and \( b = 6 \).

Step 3: Apply the Difference of Squares Formula

Apply the formula for the difference of squares: \((x - 6)(x + 6) = x^2 - 36\). This expands our equation to \( x^2 - 36 = 0 \).

Step 4: Arrange the Equation

The expression obtained is \( x^2 - 36 = 0 \). We can see that the equation is already in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 0 \), and \( c = -36 \).

Key Concepts

Roots of a Quadratic

In the world of quadratic equations, the roots are essentially the solutions to the equation. These are the values that make the quadratic expression equal zero. In a quadratic equation like \( ax^2 + bx + c = 0 \), the roots are the values of \( x \) that satisfy the equation. In simple terms, if you plug the root values into the equation in place of \( x \), the entire equation balances out, becoming zero.
For our exercise, we are given the roots of the quadratic equation as 6 and -6. This means when \( x = 6 \) and \( x = -6 \), the quadratic equation will equal zero.

• When converting roots to an equation, we use the format \((x - p)(x - q) = 0\), where \( p \) and \( q \) are the roots.
• For these roots, it becomes \((x - 6)(x + 6) = 0\).

Identifying the roots is the first step in writing the quadratic equation.

Standard Form of a Quadratic Equation

Once we have identified the format of the equation using the roots, it's important to express this equation in the standard form. Quadratic equations are often easiest to work with when they're written as \( ax^2 + bx + c = 0 \). Breaking down this format:

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

To convert our current equation \((x - 6)(x + 6) = 0)\) to standard form, we need to expand it by multiplying the terms. This involves distributing each term in \((x - 6)\) by each term in \((x + 6)\). By applying the multiplication process, we simplify the equation to get a clearer interpretation of the coefficients \( a \), \( b \), and \( c \). In our case, the standard form is obtained directly as \( x^2 + 0x - 36 = 0 \), where \( b = 0 \) since there is no \( x \) term present. Understanding the format ensures clarity and ease when solving or graphing quadratic equations.

Difference of Squares

The difference of squares is a neat algebraic trick used when you have two terms being subtracted, each of which is a perfect square. We use this pattern to simplify expressions like \((x - 6)(x + 6)\). The formula for the difference of squares is given by \( a^2 - b^2 = (a + b)(a - b) \). This formula allows us to expand expressions quickly and efficiently.
For our quadratic expression, \( (x - 6)(x + 6) \) fits perfectly into this pattern. Here, \( a = x \) and \( b = 6 \), meaning we can directly apply the formula:

• Substitute into the formula: \( x^2 - 6^2 \).
• Simplify to get \( x^2 - 36 \).

This technique provides a fast way to solve multiplication for specific quadratic expressions, making it a powerful tool in algebra. The difference of squares makes handling quadratic equations more manageable by reducing the chances of errors during multiplication.