Problem 63 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation using any method. Find only real solutions. $$(x-1)(x+2)=1$$

Step-by-Step Solution

Answer

The real solutions to the quadratic equation are \(x = 1\) and \(x = -2\)

Step 1: Expand and Rearrange the Equation

Expand the left-hand side of the equation and move the constant term from the right-hand side to the left-hand side to get it in the standard form. The equation therefore becomes: \(x^2 + 2x - x - 2 - 1 = 0\), which simplifies to \(x^2 + x - 3 = 0\)

Step 2: Apply Quadratic Formula

Apply the quadratic formula to find the solutions which is: \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). For this equation, a = 1, b = 1 and c = -3.

Step 3: Calculate Discriminant

Calculate the value of the discriminant (b² - 4ac) to determine the nature of the solutions. Here discriminant equals to \(1² - 4 * 1 * -3 = 1 + 12 = 13\)

Step 4: Solve for the roots

With the discriminant being a positive value 13, there will be two real solutions for x. Substitute the values of a, b & the calculated discriminant into the quadratic formula to determine the solutions. The solutions are: \(x₁ = [-1 + sqrt(13)] / 2 = 1\) and \(x₂ = [-1 - sqrt(13)] / 2 = -2\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It allows you to find the roots or solutions of these equations. The quadratic formula is expressed as:

• \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]

This formula shows the solutions for \( x \) in terms of the coefficients \( a \), \( b \), and \( c \). To use it, you need to identify these coefficients in your equation and substitute them into the formula.
This method is advantageous because it works for any quadratic equation, provided that you can calculate the discriminant \( b^2 - 4ac \). This formula can yield either two distinct real solutions, one real solution, or no real solutions at all, depending on the value of the discriminant.

Discriminant

The discriminant is a crucial part of the quadratic formula and determines the nature of the solutions for a quadratic equation. It is found within the square root in the formula

• \( b^2 - 4ac \)

The value of the discriminant tells us:

• If it's positive, like in our problem where it's 13, there are two distinct real solutions.
• If it's zero, there is exactly one real solution; the graph of the quadratic touches the x-axis at one point.
• If it's negative, no real solutions exist as the quadratic doesn't intersect the x-axis, instead it has two complex solutions.

The discriminant essentially gives insight into how the parabola represented by a quadratic equation behaves relative to the x-axis.

Real Solutions

Real solutions for a quadratic equation are the values of \( x \) that satisfy the equation and are real numbers. In the context of the quadratic equation we discussed, the discriminant being a positive number 13 indicates that the equation has two real solutions. Specifically, these are the x-values where the equation equals zero and are points where the graph of the quadratic crosses the x-axis.
When you have real solutions, these are tangible values you can plot on a graph and are distinct points or intersections with the x-axis. The nature of real solutions can be:

• Two distinct real solutions: When the parabola crosses the x-axis at two points, as in our example.
• One real solution: This occurs when the parabola just touches the x-axis and is often called a repeated or double root.

In summarizing the nature of solutions, understanding real versus complex solutions is vital to correctly solving quadratic equations.