Problem 35 - A Graphical Approach to Precalculus with Limits

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, expand the left side of the equation \((x+2)(x-1)\). Use the distributive property to multiply: \((x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2\). Thus, the equation becomes: \(x^2 + x - 2 = 7x + 5\).

Step 2: Move All Terms to One Side

To solve the equation, we need to set it to zero by moving all terms to one side. Subtract \(7x + 5\) from both sides:\(x^2 + x - 2 - 7x - 5 = 0\). Simplify this to obtain:\(x^2 - 6x - 7 = 0\).

Step 3: Factor the Quadratic Equation

We need to factor the quadratic equation \(x^2 - 6x - 7 = 0\). Look for two numbers that multiply to \(-7\) and add up to \(-6\). These numbers are \(-7\) and \(1\). Thus, the equation factors to:\((x-7)(x+1) = 0\).

Step 4: Solve the Factored Equation

To find the solutions, set each factor equal to zero:1. \(x - 7 = 0\), which gives the solution \(x = 7\).2. \(x + 1 = 0\), which gives the solution \(x = -1\).

Step 5: Verify the Solutions Graphically

For verification, plot the equation \((x+2)(x-1)\) and the line \(7x+5\) on the same graph. The intersections of these graphs correspond to the solutions. You'll find they intersect at \(x = 7\) and \(x = -1\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions and solve equations. It states that multiplying a single term by a sum of terms is the same as multiplying each term individually and then adding the results. The formula for the distributive property is:

\[a(b + c) = ab + ac\]
In this exercise, we start by applying the distributive property to expand the expression \((x+2)(x-1)\). By distributing \(x+2\) across \(x-1\), the expression becomes:

• \(x \cdot x = x^2\)
• \(x \cdot -1 = -x\)
• \(2 \cdot x = 2x\)
• \(2 \cdot -1 = -2\)

Adding these terms together, we simplify the expression to \(x^2 + x - 2\). Using the distributive property effectively simplifies complex multiplication and is a stepping stone to more advanced problem-solving techniques.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable \(x\). It typically takes the form:

\[y = ax^2 + bx + c\]
In our exercise, after expanding and simplifying, we obtain the quadratic equation:\[x^2 - 6x - 7 = 0\]Quadratic equations can have one, two, or no real solutions. The solutions are often referred to as "roots." These can be found using various methods such as factoring, completing the square, or the quadratic formula. Here, we've opted for factoring to solve the equation. Recognizing the coefficients and understanding how they affect the solutions is key to mastering quadratic equations.

Graphical Verification

Graphical verification involves plotting equations on a graph to visually confirm their solutions. In our exercise, this means graphing the expressions and observing where they intersect. Usually, a graph provides a clear picture of how algebraic solutions relate to a visible representation.

For example, by plotting both sides of the original equation:

• \((x+2)(x-1)\)
• \(7x + 5\)

We can visually identify the intersection points, which represent the solutions of the equation. The intersections occur where the y-values of both expressions are equal. This graphical method is especially useful when the algebraic solution is uncertain or needs confirmation.

Factoring Method

Factoring is a popular method to solve quadratic equations by expressing the equation as a product of its factors. This involves breaking down a quadratic expression into simpler binomial expressions.

In our problem, the quadratic equation \(x^2 - 6x - 7 = 0\) was factored into:\[(x-7)(x+1) = 0\]To factor effectively, look for two numbers that multiply to the constant term (in this case, \(-7\)) and add to the linear coefficient (\(-6\)). The solutions to the factored equation are found by setting each binomial equal to zero:

• \(x - 7 = 0\)
• \(x + 1 = 0\)

Thus, \(x = 7\) and \(x = -1\) are solutions. Factoring is a straightforward method which, when applicable, provides quick and clear solutions to quadratic equations.