Problem 15 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$x^{2}+9 x=-8$$

Step-by-Step Solution

Answer

The solutions to the given quadratic equation are \(x = -1\) and \(x = -8\).

Step 1: Equation Setup

Rewrite the given equation to standard quadratic form \(ax^2+bx+c=0\). So, the equation becomes \(x^2 + 9x + 8 = 0\).

Step 2: Factor

Find two numbers that multiply to give the last term (8) and add up to the second term (9). These numbers are 1 and 8. So, factoring the equation gives \((x+1)(x+8) = 0\).

Step 3: Solve for x

Set each factor equal to zero and solve for x: \nFor \(x+1=0\), we get \(x = -1\) and for \(x+8=0\), we get \(x = -8\).

Step 4: Check the solutions

Substitute the values of x into the original equation to check if both satisfy the equation. When \(x = -1\), we get \((-1)^2 + 9(-1) + 8 = 0\), which is true. When \(x = -8\), we get \((-8)^2 + 9(-8) + 8 = 0\), which is also true. Therefore, the solutions are correct.

Key Concepts

Factoring

Factoring is a technique used to simplify quadratic equations to make them easier to solve. It involves breaking down the quadratic equation into simpler multipliers, known as factors. The process typically requires transforming the equation into its standard form, such as \( ax^2 + bx + c = 0 \), if it's not already. Then, you will find two numbers that multiply to the constant term (\( c \)) and add up to the coefficient of the linear term (\( b \)). These two numbers help in writing the equation as a product of two binomial expressions. For instance, in the equation \( x^2 + 9x + 8 = 0 \), the numbers 1 and 8 multiply to 8 (the constant) and add up to 9 (the coefficient of the linear term), allowing us to factor the equation as \( (x+1)(x+8) = 0 \). Factoring can provide a direct path to finding the equation's solutions.

Solving Quadratic Equations

Solving quadratic equations involves finding the value of \( x \) that makes the equation true. Once the equation is factored, you can solve for \( x \) by applying the zero-product property. This principle states that if the product of two numbers is zero, then at least one of the numbers must be zero. After factoring the equation \( x^2 + 9x + 8 = 0 \) into \( (x+1)(x+8) = 0 \), set each factor equal to zero and solve:

• \( x + 1 = 0 \): subtract 1 from both sides to get \( x = -1 \).
• \( x + 8 = 0 \): subtract 8 from both sides to get \( x = -8 \).

Thus, the solutions or roots of the quadratic equation are \( x = -1 \) and \( x = -8 \). These are the values of \( x \) that satisfy the original equation.

X-Intercepts

The x-intercepts of a quadratic equation are the points where the graph of the equation crosses the x-axis. These intercepts correspond to the solutions of the quadratic equation, also known as the roots. In the context of our equation \( x^2 + 9x + 8 = 0 \), the x-intercepts are \( x = -1 \) and \( x = -8 \). These values represent the points \((-1, 0)\) and \((-8, 0)\) on the graph.You can find the x-intercepts by setting the quadratic equation equal to zero and solving for \( x \), which we achieve through the factoring method outlined previously. Moreover, visualizing these x-intercepts using a graphing utility can provide an additional layer of confirmation. When you graph \( y = x^2 + 9x + 8 \), you should see the curve crossing the x-axis at \( x = -1 \) and \( x = -8 \).

Checking Solutions

Checking the solutions of a quadratic equation ensures that the found values satisfy the original equation. This can be done by substituting the solutions back into the equation or using graphical verification. Let's verify the solutions \( x = -1 \) and \( x = -8 \) for the equation \( x^2 + 9x + 8 = 0 \).

• For \( x = -1 \), substitute into the equation: \((-1)^2 + 9(-1) + 8 = 1 - 9 + 8 = 0\). This holds true.
• For \( x = -8 \), substitute into the equation: \((-8)^2 + 9(-8) + 8 = 64 - 72 + 8 = 0\). This also holds true.

Both values satisfy the equation, confirming they are correct solutions. Using a graphing tool to plot the quadratic function and confirming the graph crosses the x-axis at these points offers further assurance of the accuracy of your solutions.