Problem 57 - Calculus: One and Several Variables

Question

Find the real roots of the equation. \(x^{2}-x-2=0\).

Step-by-Step Solution

Answer

The real roots of the equation \(x^{2}-x-2=0\) are \(x = 2\) and \(x = -1\).

Step 1: Identify the coefficients

The given equation is in the form \(ax^{2} + bx + c = 0\) where \(a\), \(b\), and \(c\) are coefficients. In this case, \(a = 1\), \(b = -1\), and \(c = -2\).

Step 2: Apply the Quadratic Formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{(b^{2}-4ac)}}{2a}\]\n Substituting \(a = 1\), \(b = -1\), \(c = -2\) in the formula, we get: \[x = \frac{-(-1) \pm \sqrt{(-1)^{2}-4(1)(-2)}}{2(1)}\]

Step 3: Simplify expressions

Simplifying the expressions inside and under the square root, we get: \[x = \frac{1 \pm \sqrt{1+8}}{2}\] \nFurther simplifying the expressions under the square root: \[x = \frac{1 \pm \sqrt{9}}{2}\] \nSolving the square root and the numerator: \[x = \frac{1 \pm 3}{2}\]

Step 4: Find solutions for \(x\)

This yields two solutions: \[ x = \frac{1+3}{2} = 2\] and \[x = \frac{1-3}{2} = -1\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula enables us to solve any quadratic equation, regardless of its complexity. Quadratic equations may not always be factored easily, but using the quadratic formula provides a straightforward method to obtain the roots. The formula is written as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation.
• The symbol \(\pm\) indicates that there may be two possible solutions.

Breaking down the formula:

• The term \(-b\) changes the sign of \(b\).
• \(\sqrt{b^2 - 4ac}\) is the discriminant, and it determines the nature of the roots.
• The whole expression is divided by \(2a\), ensuring we properly account for the quadratic nature of the equation.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true, essentially the solutions to the equation. Understanding these roots is crucial because they indicate where the parabola represented by the quadratic equation crosses the x-axis. Solving for the roots involves substituting values into the quadratic formula and performing straightforward calculations.

• The roots could be real or complex based on the discriminant \(b^2 - 4ac\).
• The two roots are typically distinct when the discriminant is positive.
• When the discriminant is zero, the equation has exactly one repeated real root.
• If the discriminant is negative, the roots become complex, meaning they have imaginary components and are not real intersections on the x-axis.

Using these solutions for \(x\), we interpret the graph and behavior of the quadratic function.

Real Roots

Real roots of a quadratic equation are the actual values of \(x\) that satisfy \(ax^2 + bx + c = 0\) and are found on the x-axis of the graph. They occur when the discriminant \(b^2 - 4ac\) is non-negative. Real roots tell us where the parabola intersects the x-axis, representing real-world solutions or physical points.In our exercise, the discriminant was calculated as:

• \(b^2 - 4ac = (-1)^2 - 4(1)(-2) = 1 + 8 = 9\)
• Because 9 is positive, the equation has two distinct real roots.

The roots here can be found using:

• \(x = \frac{1+3}{2} = 2\)
• \(x = \frac{1-3}{2} = -1\)

These real roots \(2\) and \(-1\) are where the original quadratic equation \(x^2 - x - 2 = 0\) meets the x-axis.