Problem 2 - Algebra 1: Concepts and Skills

Question

Write \(7 x^{2}=12+3 x\) in standard form. What is the leading coefficient?

Step-by-Step Solution

Answer

The leading coefficient of the equation is 7.

Step 1: Rearranging the equation

First, we rearrange the given equation: \(7 x^{2}- 3 x - 12 = 0\). This is accomplished by subtracting \(3x\) and 12 from both sides of the equation.

Step 2: Simplifying the equation

This gives the equation in its standard form of \(ax^{2} + bx + c = 0\). Here our equation is \(7x^{2} - 3x - 12 = 0\).

Step 3: Identifying the leading coefficient

Now we can identify the leading coefficient of the equation. The leading coefficient is the coefficient of the \(x^{2}\) term. In this case, the leading coefficient 'a' is 7.

Key Concepts

Standard Form

A quadratic equation is an equation that involves the square of an unknown variable, usually represented as \(x\). The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\). This is a critical concept as it provides a uniform way to present and solve these types of equations.

To express any quadratic equation in standard form, you'll need to rearrange it so that all terms are on one side of the equation, with zero on the other. This typically involves moving terms by adding or subtracting them to both sides.

For instance, the equation \(7x^2 = 12 + 3x\) is not in standard form initially. To convert it, we subtract \(3x\) and 12 from both sides to consolidate it into \(7x^2 - 3x - 12 = 0\). Now, it neatly fits into the \(ax^2 + bx + c = 0\) structure, which makes it easier to analyze and solve.

Leading Coefficient

In the standard form of a quadratic equation \(ax^2 + bx + c = 0\), the leading coefficient is a critical aspect. It is the coefficient \(a\) associated with the \(x^2\) term.

Understanding this coefficient is important because it can tell you a lot about the parabola when the equation is graphed. For example, if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

In our example, from the equation \(7x^2 - 3x - 12 = 0\), the leading coefficient \(a\) is clearly 7. This means our quadratic function opens upwards, and the value also influences the "width" of the parabola. A larger absolute value of \(a\) means the parabola is narrower, while a smaller absolute value of \(a\) makes it wider.

Coefficients in Algebra

Coefficients are the numerical parts of terms in an algebraic expression or equation, and they play a foundational role in manipulating and understanding algebraic equations.

In the quadratic equation \(ax^2 + bx + c = 0\), coefficients include:\

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• \(a\), the leading coefficient, responsible for the \(x^2\) term.
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• \(b\), which is the coefficient of the \(x\) term.
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• \(c\), known as the constant term or the free coefficient.
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Understanding these coefficients allows us to analyze equations more deeply. For instance, they help in factoring, determining the axis of symmetry of the parabola, and finding the roots using methods like the quadratic formula.

Recognizing how these coefficients work together can also aid in predicting the graph's behavior and solving the equation by various methods.