Problem 35 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+2)(x-1)=7 x+5$$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -1\).

Step 1: Expand the Left Side

First, we need to expand the left side of the equation \((x+2)(x-1)\). Apply the distributive property:\[(x+2)(x-1) = x(x) + x(-1) + 2(x) + 2(-1) = x^2 - x + 2x - 2\].Combine like terms:\[x^2 + x - 2\]. This simplifies the equation to \(x^2 + x - 2 = 7x + 5\).

Step 2: Move All Terms to One Side

To make the equation easier to solve, we'll subtract \(7x\) and \(5\) from both sides to set the equation to zero:\[x^2 + x - 2 - 7x - 5 = 0\].Simplify the terms to get:\[x^2 - 6x - 7 = 0\].

Step 3: Factor the Quadratic Equation

We now need to factor the quadratic equation \(x^2 - 6x - 7 = 0\). The factors of \(-7\) that add to \(-6\) are \(-7\) and \(1\).Thus, the equation factors as:\((x - 7)(x + 1) = 0\).

Step 4: Solve for x

Set each factor equal to zero to find the solutions for \(x\):1. \(x - 7 = 0\) leads to \(x = 7\).2. \(x + 1 = 0\) leads to \(x = -1\).

Step 5: Verify Graphically

Sketch the graph of two functions: \(y = (x+2)(x-1)\) and \(y = 7x+5\).- The quadratic \(y = (x+2)(x-1)\) will be a parabola opening upwards.- The linear function \(y = 7x+5\) will be a straight line.Check where these two graphs intersect, which should be at the solutions \(x = 7\) and \(x = -1\).

Key Concepts

Factoring Quadratics

Factoring quadratic equations is a powerful method for finding solutions or "roots" of these equations. A quadratic equation is any equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The goal of factoring is to rewrite the quadratic as a product of two binomials:

• Identify a pair of numbers that multiply to \(c\) (the constant term) and add to \(b\) (the linear coefficient).
• Write the quadratic expression as a product of two binomials using these numbers.

For example, in the equation \(x^2 - 6x - 7 = 0\), we look for two numbers that multiply to \(-7\) and add to \(-6\), which are \(-7\) and \(1\). Therefore, the factored form is \((x - 7)(x + 1) = 0\). Solving the factors gives us the roots of the equation: \(x = 7\) and \(x = -1\). Factoring is a method often used because it typically yields solutions directly and is quicker than completing the square or using the quadratic formula.

Graphical Solutions

Graphical solutions offer a visual representation of where two functions intersect, which represent the solutions to the equation. In our specific example,

• The quadratic equation \((x+2)(x-1)\) when expanded forms the parabola \(y = x^2 + x - 2\).
• The linear equation \(y = 7x + 5\) creates a straight line when graphed.

By plotting these two functions on a coordinate graph, you can see where they intersect, providing a graphical solution. Intersection points correspond to the values of \(x\) where both equations are equal. For this exercise, the intersections occur at \(x = 7\) and \(x = -1\), confirming our algebraic solutions. Graphical solutions are beneficial because they provide a visual check, offering insight into the behavior of the equations beyond just their solutions.

Distributive Property

The distributive property is a fundamental algebraic concept used to multiply a single term across terms within parentheses. It states that for any numbers or expressions \(a\), \(b\), and \(c\), the equation \(a(b+c) = ab + ac\) holds true. In quadratic equations, the distributive property helps expand expressions.

• Starting with \((x+2)(x-1)\), the property allows you to systematically multiply each term, resulting in \(x(x) + x(-1) + 2(x) + 2(-1)\).
• This expansion process step-by-step leads to \(x^2 - x + 2x - 2\), simplifying to \(x^2 + x - 2\).

By using the distributive property in this manner, you simplify complex expressions and lay the groundwork for further solving or factoring. This step is crucial for transforming polynomial expressions into forms that are more manageable for analysis and solution finding.