Problem 89 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ x^{2}-2 x=1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 1 - \sqrt{2}\) and \(x = 1 + \sqrt{2}\).

Step 1: Identify coefficients

From the equation \(x^2 - 2x - 1 = 0\), we can identify the coefficients as \(a = 1\), \(b = -2\), \(c = -1\).

Step 2: Calculate the discriminant

Before using the quadratic formula let's first compute the discriminant using the formula \(D = b^2 - 4ac\). Substituting the coefficient values, we get \(D = (-2)^2 - 4*1*(-1) = 4 + 4 = 8\).

Step 3: Solve for x

Now we can solve for \(x\) using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\). Let's substitute the values of \(a\), \(b\) and \(D\) into the formula: \(x = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}\).

Key Concepts

Quadratic Formula

When solving quadratic equations, the quadratic formula is a powerful tool that provides a solution to equations of the form \( ax^2 + bx + c = 0 \). It encapsulates the process of finding the roots of the equation in a neat, single expression: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

To apply this formula correctly, you first need to ensure your equation is arranged in the standard form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients. The formula then tells us that by substituting these coefficients into it, we can directly find the variable \(x\), which are the points where the quadratic equation intersects the \(x\)-axis, also known as the roots of the equation. This direct approach bypasses the need for factoring the quadratic, which can be particularly useful when the equation does not factor nicely.

Discriminant Calculation

The discriminant in a quadratic equation is found within the square root of the quadratic formula and is denoted as \( D \). The formula for the discriminant is \( D = b^2 - 4ac \). It plays a crucial role in predicting the nature of the roots without actually solving the equation.

Understanding the Value of the Discriminant

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), the roots are complex and come in conjugate pairs.

Calculating the discriminant gives us immediate insight into the solutions of the quadratic equation; for example, a positive discriminant in our original equation led to the roots \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\), indicating two distinct real solutions.

Coefficient Identification

The correct identification of coefficients in a quadratic equation is the foundational step for both the discriminant calculation and the application of the quadratic formula. Coefficients \(a\), \(b\), and \(c\) represent specific values in the standard quadratic equation, \(ax^2 + bx + c = 0\).

Identifying Coefficients:

\(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.

• The coefficient \(a\) influences the parabola's opening direction and its steepness.
• The coefficient \(b\) affects the position of the parabola along the \(x\)-axis.
• The constant term \(c\) dictates where the parabola crosses the \(y\)-axis.

In our example, \(a = 1\), \(b = -2\), and \(c = -1\) were quickly identified, setting the stage for further steps in solving the equation. Accurately pinpointing these coefficients is key to ensuring that the subsequent use of the discriminant and the quadratic formula yields correct solutions.