Problem 26 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 0=x^{2}+6 x-1 $$

Step-by-Step Solution

Answer

Answer: In the given quadratic equation, the values are \(a = 1\), \(b = 6\), and \(c = -1\).

Step 1: Identify the general form of a quadratic equation

The general form of a quadratic equation is given by: $$ ax^2 + bx + c = 0 $$ Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

Step 2: Compare the given equation to the general form

The given equation is: $$ 0 = x^2 + 6x - 1 $$ Now, let's rewrite the given equation by rearranging the terms to match the general form (\(ax^2 + bx + c = 0\)): $$ x^2 + 6x - 1 = 0 $$

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

Looking at the equation \(x^2 + 6x - 1 = 0\), we can now identify the values of \(a\), \(b\), and \(c\): - \(a\) is the coefficient of the \(x^2\) term, which is \(1\) (since no other number is in front of the \(x^2\) term, it is assumed to be \(1\)) - \(b\) is the coefficient of the \(x\) term, which is \(6\) - \(c\) is the constant term, which is \(-1\) So, the values of \(a\), \(b\), and \(c\) in the given quadratic equation are \(a = 1\), \(b = 6\), and \(c = -1\).

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra that students need to master. Quadratic equations are of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the unknown variable. To solve a quadratic equation, one must find the values of \(x\) that make the equation true.

There are several methods to solve quadratic equations, including factoring, using the quadratic formula, completing the square, or graphing. Most students use the quadratic formula as it works for any quadratic equation, provided that \(a\), \(b\), and \(c\) are known. When coefficients are fractions or if the equation is complex, simplifying the equation first can often make the process easier.

It's essential for students to practice solving quadratic equations using different methods, so they become comfortable with the processes and know when to use each method effectively. Remember, practicing with various coefficients and forms of quadratic equations will build confidence and skill in this area.

Coefficients in Quadratic Equations

In the context of quadratic equations, understanding coefficients is critical. The coefficients are the numerical factors that multiply the variables. In a standard quadratic equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are the coefficients with distinct roles:

• \(a\) is the coefficient of \(x^2\) and dictates the curvature of the parabola formed by the quadratic equation when graphed.
• \(b\) is the coefficient of \(x\) and influences how the parabola shifts and opens left or right.
• \(c\) is the constant term and affects the vertical shift of the parabola's vertex.

Remember, if \(a\) is zero, the equation is not quadratic but linear. Students should note these differences as they greatly assist in the visualization of the graph of a quadratic equation and in understanding the impact each coefficient has on the equation’s solutions.

Quadratic Formula

The quadratic formula is one of the most reliable tools for solving quadratic equations and is derived from the process of completing the square. For a quadratic equation in the form \(ax^2 + bx + c = 0\), the solutions for \(x\) can be found using the formula:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]

To apply the quadratic formula, identify the coefficients \(a\), \(b\), and \(c\), then simply plug them into the formula. The \(\pm\) symbol denotes that there are generally two possible solutions, or roots, for \(x\). The term \(b^2 - 4ac\) is called the discriminant and it determines the nature of the roots; if it is positive, there are two real and distinct solutions; if it's zero, there is just one real solution. However, if the discriminant is negative, the equation has two complex solutions.

Using the quadratic formula is especially handy when the equation is difficult to factor or when exact solutions are required. Students should familiarize themselves with the formula and practice using it with different quadratic equations to become proficient at finding solutions quickly and accurately.