Problem 79 - College Algebra

Question

Solve each equation in Exercises \(73-98\) by the method of your choice. \(x^{2}-2 x=1\)

Step-by-Step Solution

Answer

Therefore, the roots of the equation \(x^{2}-2x=1\) are \(x=1+\sqrt{2}\) and \(x=1-\sqrt{2}\).

Step 1: Rewrite the Equation

To make it easier to solve this quadratic equation, one must first rewrite the equation as \(x^{2}-2x-1=0\) by subtracting 1 from both sides of the original equation. So, the task is to find values of x for which \(x^{2}-2x-1=0\).

Step 2: Calculate Discriminant

The discriminant is used in order to determine the type of roots the quadratic equation will have. Find the discriminant using the formula \(D=b^{2}-4ac\), where \(a, b, c\) are the coefficients of the equation. In the equation \(x^{2}-2x-1=0\), \(a=1, b=-2, c=-1\). Hence, \(D=(-2)^{2}-4(1)(-1)=4+4=8\). The discriminant \(D\) is positive, so the equation will have two distinct real roots.

Step 3: Calculate the Roots

Compute the two roots using the quadratic formula which states that if we have a quadratic equation of form \(ax^{2} + bx + c=0\), then its roots are given by \(x=\frac{-b\pm \sqrt{D}}{2a}\). Substituting \(a=1\), \(b=-2\), and \(D=8\) into the formula gives \(x=\frac{2\pm \sqrt{8}}{2}\). Simplify this to find the roots \(x=1+\sqrt{2}\) and \(x=1-\sqrt{2}\).

Key Concepts

Discriminant

The discriminant is a key concept in understanding quadratic equations, as it informs us about the nature of the roots. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is found using the formula:\[D = b^2 - 4ac\] The coefficients \(a, b, \text{and } c\) come directly from the quadratic equation. In this scenario, \(a = 1\), \(b = -2\), and \(c = -1\). By plugging these values into the formula, the discriminant is:\[D = (-2)^2 - 4(1)(-1) = 4 + 4 = 8\] When the discriminant is positive, it indicates that the quadratic equation has two distinct real roots. A positive discriminant, such as in our example, generally tells us that the parabola represented by the quadratic equation intersects the x-axis at two points. Understanding the value of the discriminant helps predict the number of real roots and their nature, which can be real or imaginary, depending on whether the discriminant is positive, zero, or negative.

Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. This tool is optimal when the equation cannot be factored easily. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{D}}{2a}\] Here, \(b\) is the coefficient of \(x\), \(a\) is the coefficient of \(x^2\), and \(D\) is the discriminant. In our quadratic equation \(x^2 - 2x - 1 = 0\), we have:\[- b = -(-2) = 2\]Using \(a = 1\) and previously calculated \(D = 8\), we substitute these values into the formula:\[x = \frac{2 \pm \sqrt{8}}{2}\] Further simplifying, you obtain:\[x = 1 \pm \sqrt{2}\]This results in two solutions: \(x = 1 + \sqrt{2}\) and \(x = 1 - \sqrt{2}\). The "\(\pm\)" symbol in the formula allows us to find both solutions, by considering both addition and subtraction of the square root value. The quadratic formula guarantees solutions for any quadratic equation, even when other methods like factoring are not applicable.

Real Roots

Real roots refer to the values of \(x\) that satisfy the quadratic equation. These roots are solutions where the equation equals zero. Given that the discriminant \(D = 8\) is positive, it indicates that our equation has two distinct real roots.To understand this better:

• When \(D > 0\), you have two distinct real roots.
• When \(D = 0\), you have exactly one real root, also known as a repeated or double root.
• When \(D < 0\), the equation has no real roots, and the solutions are complex or imaginary.

Real roots are the x-values at which the parabola formed by the quadratic equation crosses the x-axis. In our example, solving using the quadratic formula gives us the real roots \(x = 1 + \sqrt{2}\) and \(x = 1 - \sqrt{2}\). These points represent where the graph of the quadratic equation intersects the horizontal line of the x-axis. Recognizing real roots is crucial for understanding how quadratic functions relate to their graphs.