Problem 1 - Algebra 2

Question

Write a quadratic equation with the given root(s). Write the equation in standard form. \(-4,7\)

Step-by-Step Solution

Answer

The quadratic equation is \( x^2 - 3x - 28 = 0 \).

Step 1: Understanding the Roots

We are given the roots of the quadratic equation: \(-4\) and \(7\). A quadratic equation with roots \(r_1\) and \(r_2\) can be written in the form:\[ (x - r_1)(x - r_2) = 0 \] Plugging in our roots, the equation becomes:\[ (x + 4)(x - 7) = 0 \]

Step 2: Expand the Equation

To write the equation in standard form, we need to expand the expression:\[ (x + 4)(x - 7) \]Use the distributive property (FOIL method):1. First: \( x \times x = x^2 \)2. Outer: \( x \times -7 = -7x \)3. Inner: \( 4 \times x = 4x \)4. Last: \( 4 \times -7 = -28 \)Putting it all together:\[ x^2 - 7x + 4x - 28 \]

Step 3: Simplify and Write in Standard Form

Simplify the equation from the previous step by combining like terms:\( -7x + 4x = -3x \)Thus the equation becomes:\[ x^2 - 3x - 28 = 0 \]This is the quadratic equation in standard form \( ax^2 + bx + c = 0 \).

Key Concepts

Roots of a Quadratic Equation

When we talk about the 'roots of a quadratic equation’, we are referring to the values that make the equation equal to zero. These values are solutions to the equation. Consider a quadratic equation given by the general form,\[ ax^2 + bx + c = 0 \]The roots of this equation can be defined as the values of \( x \) for which the equation holds true, meaning the expression results in zero. These roots are frequently termed as \( r_1 \) and \( r_2 \). In our exercise, the given roots are \(-4\) and \(7\). With these values, the quadratic equation can be derived by setting up the expression:\[(x - r_1)(x - r_2) = 0\]which accounts for the property that any number multiplied by zero will result in zero. It is an elegant way to construct the quadratic equation starting from its solutions.

Standard Form

The 'standard form' of a quadratic equation is a well-organized way of writing it, typically represented as:\[ ax^2 + bx + c = 0 \]This form serves several purposes, like making it easier to identify the coefficients \(a\), \(b\), and \(c\). These coefficients play a vital role in many mathematical operations, such as factoring, solving with the quadratic formula, and graphing the parabola associated with the equation. To convert an equation from product form, such as \((x + 4)(x - 7) = 0\), to standard form, we must expand using operations like the distributive property. Our end goal is an equation that starts with \(x^2\) and includes all terms down to a constant value, perfectly aligning with the standard form structure.

Distributive Property

The 'distributive property' is a pivotal mathematical rule which allows us to simplify expressions. It talks about how to break down multiplication distributed over addition or subtraction, and is expressed as:\[ a(b + c) = ab + ac \]In the context of our exercise, we use the distributive property to expand \((x + 4)(x - 7)\). By applying the famous FOIL method—which stands for First, Outer, Inner, Last—we systematically multiply the terms:

• First: \(x\) times \(x\) gives \(x^2\)
• Outer: \(x\) times \(-7\) gives \(-7x\)
• Inner: \(4\) times \(x\) gives \(4x\)
• Last: \(4\) times \(-7\) gives \(-28\)

Combining these results gives us the expanded form \(x^2 - 7x + 4x - 28\), which upon simplifying by adding like terms results in \(x^2 - 3x - 28\). This method brings us closer to the standard form and neatly demonstrates the power of distribution in algebraic operations.