Problem 60 - Elementary Algebra

Question

\(k^{2}+6 k+5=0\)

Step-by-Step Solution

Answer

Solutions are \(k = -1\) and \(k = -5\).

Step 1: Identify the quadratic equation

Recognize that the given equation is a quadratic equation of the form oindent\(ax^{2} + bx + c = 0\). The equation is \(k^{2} + 6k + 5 = 0\), where \(a = 1\), \(b = 6\), and \(c = 5\).

Step 2: Factor the quadratic equation

Look for two numbers that multiply to \(ac = 1 \cdot 5 = 5\) and add up to \(b = 6\). These numbers are 1 and 5.

Step 3: Write the equation in factored form

Express the quadratic equation as \((k + 1)(k + 5) = 0\).

Step 4: Solve for the variable

Set each factor equal to zero and solve for \(k\): \(k + 1 = 0\) or \(k + 5 = 0\). Solving these gives:- \(k = -1\)- \(k = -5\)

Key Concepts

factoring quadratics

Factoring quadratics is a method used to solve quadratic equations. A quadratic equation is any equation of the form \(ax^{2} + bx + c = 0\), where \(a, b,\) and \(c\) are constants.
Factoring involves rewriting the quadratic equation as a product of two binomials.
For example, in the equation \(k^{2} + 6k + 5 = 0\), we look for two numbers that multiply to \(ac\) (the product of the first and last coefficients) and add up to \(b\) (the middle coefficient).
Here, those numbers are 1 and 5, since \(1 \times 5 = 5 \) and \(1 + 5 = 6\).
We rewrite the equation as \((k + 1)(k + 5) = 0\).
Factoring makes it easy to find the solutions to the equation.

roots of equations

The roots of an equation are the values of the variable that make the equation true.
For a quadratic equation like \(k^{2} + 6k + 5 = 0\), finding the roots means finding the values of \(k\) that satisfy the equation.
After factoring the quadratic equation into \((k + 1)(k + 5) = 0\), we can easily determine the roots.
We solve for when each binomial equals zero: \(k + 1 = 0\) or \(k + 5 = 0\).
Solving these simple equations gives us the roots: \(k = -1\) and \(k = -5\).
So, the roots of the quadratic equation \(k^{2} + 6k + 5 = 0\) are \(k = -1\) and \(k = -5\).

zero product property

The zero product property states that if the product of two numbers is zero, then at least one of the numbers must be zero.
This property is crucial when solving factored quadratic equations.
In our example, after factoring \(k^{2} + 6k + 5 = 0\) into \((k + 1)(k + 5) = 0\), the zero product property allows us to set each factor equal to zero.
We then find \(k + 1 = 0 \) and \(k + 5 = 0\).
Solving these, we have \(k = -1\) and \(k = -5\).
Therefore, the zero product property helps us determine the roots of the equation by breaking it down into simpler parts.

identifying coefficients

Identifying the coefficients of a quadratic equation is an essential step in solving it.
The standard form of a quadratic equation is \(ax^{2} + bx + c = 0\), where \(a, b,\) and \(c\) are coefficients.
In the equation \(k^{2} + 6k + 5 = 0\), we identify \(a = 1\) (the coefficient of \(k^{2}\)), \(b = 6\) (the coefficient of \(k\)), and \(c = 5\) (the constant term).
These coefficients help us determine how to factor the quadratic equation.
We use them to find two numbers that multiply to \(ac\) and add to \(b\), making it easier to write the equation in its factored form.
Properly identifying coefficients is the first step in many solution methods for quadratic equations.