Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ -x^{2}-7 x=10 $$

Step-by-Step Solution

Answer

The two solutions for the equation are x=-2 and x=-5.

Step 1: Arrange into standard form

Start by arranging the equation into the standard form. - Add 10 to both sides of the equation to isolate the terms with x on the left side and the constant on the right: \(x^{2} + 7x + 10 = 0 \)

Step 2: Factor

Next, factor the left side of the equation. This can be done by finding two numbers that multiply to 10 and add to 7, which are 2 and 5. Therefore, the factored form is: \((x + 2)(x + 5) = 0 \)

Step 3: Solve

To solve for the roots, set each factor equal to zero and solve for x. The solution will be \(x=-2\) and \(x=-5\)

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental technique used to solve quadratic equations. When we factor a quadratic, we express it as a product of two binomials. This expresses the quadratic equation in a simpler form, making it easier to find solutions. For an equation like \(x^2 + 7x + 10\), we look for two numbers that multiply to give the constant term (here, 10) and add up to the linear coefficient (here, 7). In this case, the numbers 2 and 5 satisfy these conditions. Thus, the equation can be factored as \((x + 2)(x + 5)\). This method is efficient when the quadratic can easily be expressed as a product of binomials with integer coefficients. Some tips for successful factoring include:

• Always look for a greatest common factor first.
• Check the signs: they depend on the signs of the terms in the original quadratic.
• Verify your factorization by expanding it again to check if you get the original quadratic back.

Standard Form

Quadratic equations can appear in various forms, but the standard form is \(ax^2 + bx + c = 0\). In this format, \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. Converting a quadratic equation to standard form involves rearranging the terms so that the equation equals zero. For example, consider the equation \(-x^2 - 7x = 10\). To transform it into standard form, we move all terms to one side to get \(x^2 + 7x + 10 = 0\). This rearrangement is crucial as it prepares the equation for methods like factoring.When shifting terms:

• Ensure that you perform the same operation on both sides of the equation.
• Simplify to combine like terms, if any.
• Check if the equation has a perfect square or can be factored directly.

Roots of an Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These values are also known as solutions or zeroes. For a factored form such as \((x + 2)(x + 5) = 0\), finding the roots involves setting each parenthesis equal to zero: \(x + 2 = 0\) and \(x + 5 = 0\).Solving these results in roots \(x = -2\) and \(x = -5\). These are the points where the graph of the quadratic function meets the x-axis. To summarize, the roots:

• Are found by setting each factor equal to zero from the factored form.
• Represent the solutions to the original quadratic equation.
• Indicate where the function crosses or touches the x-axis on a graph.

Understanding the roots helps in understanding the graph of the quadratic equation, particularly its intersections with the x-axis.