Problem 75 - Algebra A Combined Function

Question

For each quadratic equation, choose the correct substitution for \(a, b,\) and \(c\) in the standard form \(a x^{2}+b x+c=0 .\) \(x^{2}=-10\) a. \(a=1, b=0, c=-10\) b. \(a=1, b=0, c=10\) c. \(a=0, b=1, c=-10\) d. \(a=1, b=1, c=10\)

Step-by-Step Solution

Answer

Option b: \( a=1, b=0, c=10 \) is correct.

Step 1: Understand the Standard Form of Quadratic Equation

The standard form of a quadratic equation is given by \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants. The term \( ax^2 \) must be present, meaning \( a eq 0 \). The equation includes terms with \( x^2 \), \( x \), and a constant.

Step 2: Identify Terms from Given Equation

The given equation is \( x^2 = -10 \). To compare it with the standard form, it needs to be written as \( x^2 + 0x + 10 = 0 \). Here, \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

Step 3: Substitute Values of a, b, and c

By comparing \( x^2 = -10 \) rewritten as \( x^2 + 0x + 10 = 0 \) with the standard form, assign values: \( a = 1 \), \( b = 0 \), \( c = 10 \). This matches when \( x^2 = -10 \) is rearranged to fit the quadratic format.

Step 4: Choose the Correct Option

From the substitutions, option b: \( a=1, b=0, c=10 \) matches the identified values from the equation \( x^2 + 0x + 10 = 0 \).

Key Concepts

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is written as \( ax^2 + bx + c = 0 \). This is a way of organizing the terms of a quadratic equation, making it easier to identify the components. Quadratic equations have a term with \( x^2 \) (which is always present), a term with \( x \), and a constant term.
All these need to align with a particular structure:

• \( a \) is the coefficient of the \( x^2 \) term.
• \( b \) is the coefficient of the \( x \) term.
• \( c \) is the constant term without any variable attached to it.

The most important element in a quadratic equation is the term \( ax^2 \), because if \( a = 0 \), it would no longer be quadratic, but rather linear. Recognizing this standard form helps in easily solving and manipulating quadratic equations.

Substitution Method

The substitution method is a straightforward approach used to simplify and solve problems like quadratic equations. It involves replacing variables with specific values or expressions to make the problem easier to solve.
In the context of quadratic equations, you substitute the known terms into the standard form to identify unknown coefficients or constants.

• Substitute known values to directly identify \(a\), \(b\), and \(c\).
• Ensure that the substitution aligns the given equation with the standard format \(ax^2 + bx + c = 0\).

This method supports understanding by turning abstract equations into manageable numbers that are easier to handle, making it a foundational step in solving quadratic equations.

Coefficients Identification

Identifying coefficients is a key step in working with the standard form of a quadratic equation. Each coefficient represents a specific part of the equation. By reorganizing the equation to fit the standard form, you can easily pinpoint these values:

• The coefficient \(a\) determines the curvature of the parabola represented by the equation.
• The coefficient \(b\) influences the direction or slant of the parabola.
• The constant \(c\) dictates where the graph of the equation crosses the y-axis.

In the problem given, coefficients are matched to compare the equation with the standard form. For \(x^2 = -10\), you rearrange it as \(x^2 + 0x + 10 = 0\) and thus identify \(a = 1\), \(b = 0\), and \(c = 10\). Recognizing these coefficients is crucial for further manipulation and solution of the equation, including graphing and factoring.

Constant Term in Quadratic Equations

The constant term in a quadratic equation, generally denoted as \(c\), is the term without any variable attached. It's a fixed number that shifts the entire equation up or down on the graph.
The role of \(c\) is often understated, but it's significant:

• It helps determine the position of the parabola along the y-axis.
• A positive \(c\) shifts the graph upwards, while a negative \(c\) shifts it downwards.
• The value of \(c\) can be directly observed from how the equation is structured when rewritten to fit the standard form.

In the given problem, the equation \(x^2 = -10\) becomes \(x^2 + 0x + 10 = 0\), identifying \(c = 10\). This value is significant because it relates directly to how the equation would graphically intersect the y-axis, giving essential context in understanding quadratic functions.