Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+4)(x-1)=-5 x-4$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -8\).

Step 1: Expand the Left Side

The left side of the equation is \((x+4)(x-1)\). Expand it using the distributive property: \((x+4)(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4.\)

Step 2: Set the Equation to Zero

Rearrange the equation to \(x^2 + 3x - 4 + 5x + 4 = 0.\)Simplify the expression to get:\[x^2 + 8x = 0.\]

Step 3: Factor the Quadratic Equation

Factor the quadratic equation \(x^2 + 8x = 0\) by taking out the common factor of \(x\):\(x(x + 8) = 0.\)

Step 4: Solve for x

Using the zero-product property, set each factor equal to zero and solve:\(x = 0\) or \(x + 8 = 0\).So, \(x = 0\) or \(x = -8\).

Step 5: Graphical Solution Verification

To verify the solutions graphically, plot the original functions \((x+4)(x-1)\) and \(-5x-4\):- The function \((x+4)(x-1)\) is a parabola opening upwards.- The function \(-5x-4\) is a straight line.Check for intersection points and verify they correspond to \(x = 0\) and \(x = -8\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a method used to simplify expressions by expressing them as a product of their factors. In the context of quadratic equations, factoring is a key step in solving them. Consider the equation \(x^2 + 8x = 0\). To factor it, look for common factors in each term. Here, both terms contain \(x\), so, you can factor out \(x\):

• Identify common factors: In \(x^2 + 8x\), the common factor is \(x\).
• Write the equation as a product: \(x(x + 8) = 0\). This expresses the quadratic equation as two separate factors.

By factoring, you transform the original quadratic into a product of linear terms, which is much easier to work with. This step of breaking down equations into their factors simplifies the problem and lays the foundation for applying the zero-product property.

Zero-Product Property

The zero-product property is a fundamental principle used to solve equations that have been factored into a product of terms. According to this property, if the product of two or more terms equals zero, then at least one of the terms must be zero. This is crucial in solving equations like \(x(x + 8) = 0\).

• Apply zero-product property: Either \(x = 0\) or \(x + 8 = 0\).
• Solve for \(x\): Setting \(x = 0\) gives you the solution \(x = 0\), and setting \(x + 8 = 0\) gives you the solution \(x = -8\).

Using this property allows us to find solutions to the equation, as seen in this example. By setting each factor to zero and solving, we determine the values of \(x\) that satisfy the original equation. This technique is handy for solving polynomial equations that have been properly factored.

Distributive Property

The distributive property is an essential algebraic rule that allows for breaking down complex expressions. This rule states that \(a(b + c) = ab + ac\), which helps in expanding or simplifying expressions. For instance, in the exercise, we used the distributive property to expand \((x+4)(x-1)\):

• Expand each term: Apply \((x+4)(x-1) = x^2 - x + 4x - 4\).
• Combine like terms: Combine \(-x + 4x\) to get \(x^2 + 3x - 4\).

The distributive property helped transform a factored expression into a standard quadratic equation form which is easier to manipulate. Understanding this property is vital for both simplifying expressions and establishing equations in a form suitable for applying further algebraic techniques like factoring.

Graphical Solutions

Graphical solutions involve visual representation to verify or find solutions to equations. To understand the solution of the quadratic equation \((x+4)(x-1) = -5x-4\), we can graph both sides of the equation:

• Graph \((x+4)(x-1)\): This is an upward-opening parabola.
• Graph \(-5x-4\): This is a straight line.

Plotting these functions on the same graph and identifying their intersection points helps confirm the algebraic solutions. The intersection points represent the values of \(x\) that satisfy the equation. Here, the graphical method confirms the solutions \(x = 0\) and \(x = -8\), providing a visual validation of the analytical approach.