Problem 32 - Elementary Algebra

Question

Identify the coefficients \(a, b,\) and \(c\) used in the quadratic formula. Do not solve. $$ -x 2+5 x-14=0 $$

Step-by-Step Solution

Answer

The coefficients are \(a = -1\), \(b = 5\), \(c = -14\).

Step 1: Standard Form of Quadratic Equation

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In this form, \(a\), \(b\), and \(c\) are coefficients that we need to identify.

Step 2: Write the Given Equation in Standard Form

The given equation is \(-x^2 + 5x - 14 = 0\). Write it as \(-1x^2 + 5x - 14 = 0 \) to clearly see the coefficients.

Step 3: Identify Coefficient a

The coefficient \(a\) corresponds to the term \(x^2\). In \(-1x^2\), \(a = -1\).

Step 4: Identify Coefficient b

The coefficient \(b\) corresponds to the term \(x\). In \(5x\), \(b = 5\).

Step 5: Identify Coefficient c

The coefficient \(c\) is the constant term. In the equation, \(c = -14\).

Key Concepts

Coefficients

In a quadratic equation, coefficients play a crucial role. They are the numerical values that accompany variables in the equation. In our specific quadratic equation, \(-x^2 + 5x - 14 = 0\), we need to identify the coefficients for each term. Breaking it down:

• **Coefficient \(a\)**: This is the number attached to \(x^2\). In our example, we have \(-1x^2\), hence \(a = -1\).
• **Coefficient \(b\)**: This accompanies the \(x\) term. Here, it is \(5x\), therefore \(b = 5\).
• **Coefficient \(c\)**: This is the constant term without any \(x\) attached. In our equation, it is \(-14\), so \(c = -14\).

Identifying these coefficients is essential for further operations, especially when using the quadratic formula to find equation roots.

Standard Form

The standard form of a quadratic equation is key to analyzing and solving it effectively. It is expressed as \(ax^2 + bx + c = 0\). This form helps clarify the role of each term:

• **\(ax^2\)**: Represents the quadratic term which shapes the parabola by defining its direction and width.
• **\(bx\)**: The linear term impacting the skew or tilt of the parabola.
• **\(c\)**: The constant term which moves the entire graph up or down.

Writing our given equation, \(-x^2 + 5x - 14 = 0\), in this standard form makes it easier to identify the coefficients \(a = -1\), \(b = 5\), and \(c = -14\). This format is also beneficial when inputting values into the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. This formula is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]It utilizes the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation. Here's why it's useful:

• **Applicability**: Works for any quadratic equation, whether it can be factored or not.
• **Precision**: Provides exact values for the roots (solutions), including complex ones.
• **Consistent Results**: Because it follows a set formula, it consistently yields correct results when used properly.

Knowing the coefficients earlier allows you to confidently apply them into this formula to find the roots of your equation. For instance, with coefficients \(a = -1\), \(b = 5\), and \(c = -14\), you can insert them into the quadratic formula to solve for \(x\).