Problem 20 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 4 y^{2}=0 $$

Step-by-Step Solution

Answer

Question: Identify the values of \(a\), \(b\), and \(c\) in the quadratic equation \(4y^2=0\). Answer: The values of \(a\), \(b\), and \(c\) in the quadratic equation \(4y^2=0\) are \(a=4\), \(b=0\), and \(c=0\).

Step 1: Write the quadratic equation in standard form

To find the values of \(a\), \(b\), and \(c\), we need to rewrite the given equation \(4y^2=0\) in the standard form of a quadratic equation, which is \(ay^2+by+c=0\).

Step 2: Identify the coefficients

Observe the equation written in standard form: $$ 4y^2+0y+0=0 $$ Now we can easily identify the values for \(a\), \(b\), and \(c\) by comparing this equation to the standard form \(ay^2+by+c=0\).

Step 3: Write the values of \(a\), \(b\), and \(c\)

From the comparison: - \(a\) is the coefficient of the \(y^2\) term, so \(a=4\). - \(b\) is the coefficient of the \(y\) term, so \(b=0\). - \(c\) is the constant term, so \(c=0\). Thus, the values of \(a\), \(b\), and \(c\) are \(4\), \(0\), and \(0\), respectively.

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is crucial for solving problems that deal with parabolas or projectile motions. The standard form is generally written as \( ax^2 + bx + c = 0 \), where \( a \) is the coefficient of the squared term, \( b \) is the coefficient of the linear term, and \( c \) is the constant term. It's important to note that \( a \) cannot be zero because otherwise, the equation would not be quadratic but rather linear.

In the context of the sample exercise \( 4y^2 = 0 \), rewriting it to the standard form simply means transforming it to \( 4y^2 + 0y + 0 = 0 \), even though the latter two terms are invisible in the original equation, they're implied to be there. Embracing this concept aids in clearly understanding the structure and components of quadratic equations, allowing for accurate analysis and solution finding.

Coefficients of a Quadratic Equation

In the realm of algebra, identifying the coefficients of a quadratic equation is a basic yet integral skill. These coefficients, represented by \( a \) for the quadratic term, \( b \) for the linear term, and \( c \) for the constant term, play a vital role in characterizing the parabola's shape and position when graphed. Notably, the value of \( a \) determines the parabola's direction (upward if positive, downward if negative) and its width (wider for smaller absolute values of \( a \), narrower for larger absolute values).

In our example equation \( 4y^2 + 0y + 0 = 0 \), the coefficient \( a \) is 4, indicating a narrow, upward-facing parabola. Coefficient \( b \) is 0, implying no tilt to the parabola, and \( c \) is also 0, positioning the vertex of the parabola at the origin of the graph. By mastering coefficients, students can not only solve equations but also predict and graph quadratic functions effectively.

Identifying Quadratic Equation Values

The ability to identify quadratic equation values is a fundamental step in the solution process. These values are synonymous with the coefficients and constant introduced earlier. To find them, one must inspect the equation and match terms with their respective places in the standard form \( ax^2 + bx + c \).

For the equation in question, \( 4y^2 + 0y + 0 = 0 \), the term with the variable squared (\( y^2 \) in this case) comes accompanied by its coefficient \( a \)—here, it's 4. The term with the single power of the variable (\( y \) here) has the coefficient \( b \)—in the example, this coefficient is absent, hence it is 0. Finally, the last number standing alone is the constant term \( c \)—this too equates to 0. Identifying these values immediately sets the stage for further analysis like factoring, completing the square, or using the quadratic formula for finding the roots of the equation.