Problem 19 - Beginning and Intermediate Algebra

Question

Solve each equation. $$v^{2}+15 v+56=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(v^2 + 15v + 56 = 0\) are \(v = -7\) and \(v = -8\).

Step 1: Write down the given equation

We have the quadratic equation: \(v^2 + 15v + 56 = 0\)

Step 2: Factor the quadratic expression

We need to find two numbers whose product is equal to 56 and whose sum is equal to 15. These two numbers are 7 and 8. So, we can rewrite the given equation as: \((v + 7)(v + 8) = 0\)

Step 3: Set each factor equal to zero

Now, we set each factor equal to zero to find the possible values of v: \(v + 7 = 0\) or \(v + 8 = 0\)

Step 4: Solve for v in each equation

From the first equation, we get: \(v = -7\) From the second equation, we get: \(v = -8\) So, the solutions to the given equation are: \(v = -7\) and \(v = -8\)

Key Concepts

Factoring Quadratic Expressions

Quadratic expressions often take the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Factoring these expressions is a key step in solving quadratic equations. The goal of factoring is to rewrite the quadratic as a product of two binomials. In the exercise, the quadratic \(v^2 + 15v + 56\) is factored into \((v + 7)(v + 8)\).

To factor a quadratic expression, one should look for two numbers that multiply to the constant term (in this case, 56) and add up to the linear coefficient (in this case, 15). The numbers 7 and 8 multiply to give 56 and add up to 15, making them the correct pair of factors. Once these numbers are identified, the quadratic can be rewritten as the product of two binomials, making it easier to solve.

Solving Equations

Once a quadratic expression is factored, solving the equation becomes straightforward. The exercise asks us to solve \((v + 7)(v + 8) = 0\). According to the zero-product property, if the product of two numbers is zero, at least one of the numbers must be zero. This is the principle behind solving factored quadratic equations.

Applying this concept, we set each binomial equal to zero:

• \(v + 7 = 0\)
• \(v + 8 = 0\)

By solving these linear equations separately, we find \(v = -7\) and \(v = -8\). These are the solutions to the quadratic equation. Always ensure that after solving each factor individually, you check each solution by substituting back into the original equation to validate your results.

Algebraic Equations

Understanding algebraic equations is crucial when dealing with quadratic equations. An algebraic equation, like \(v^2 + 15v + 56 = 0\), is a statement of equality involving variables and constants.

Such equations can represent real-world problems and allow for logical relationships to be established between variables. The process of solving involves finding the variable values that satisfy the equation. With quadratic equations, the typical solving method includes:

• Rearranging the equation to set it to zero
• Factoring, if possible
• Using the zero-product property

Quadratic equations specifically represent a parabolic graph, with the points where the solutions occur being the roots or zeros of the equation. Mastering these algebraic processes provides a foundation for more advanced mathematical concepts.