Problem 6 - Elementary Algebra

Question

Determine whether the given set of values are solutions to the quadratic equation. $$ \\{-5,1\\} ; x 2-4 x-5=0 $$

Step-by-Step Solution

Answer

Neither \(-5\) nor \(1\) are solutions to the equation \(x^2 - 4x - 5 = 0\).

Step 1: Identify the Quadratic Equation

The quadratic equation given is \( x^2 - 4x - 5 = 0 \). We need to determine if the values \(-5\) and \(1\) are solutions to this equation.

Step 2: Test Value: \(x = -5\)

Substitute \(-5\) into the equation: \((-5)^2 - 4(-5) - 5 = 0\). Calculate: \(25 + 20 - 5 = 40\). Since 40 is not equal to 0, \(x = -5\) is not a solution.

Step 3: Test Value: \(x = 1\)

Substitute \(1\) into the equation: \(1^2 - 4(1) - 5 = 0\). Calculate: \(1 - 4 - 5 = -8\). Since -8 is not equal to 0, \(x = 1\) is not a solution.

Key Concepts

Solution Verification

Solution verification is a process used in mathematics to confirm whether a given value satisfies an equation. When dealing with quadratic equations, this involves substituting potential solutions back into the original equation to see if they hold true. In our case, we were given the quadratic equation \( x^2 - 4x - 5 = 0 \) and needed to verify if the values \(-5\) and \(1\) are actual solutions.

• Substitute the value of \(x\) into the equation.
• Solving the equation should yield a result equalling zero for it to be a solution.

As we saw, when verifying \(x = -5\), we found the equation did not result in zero (it equaled 40), and similarly, \(x = 1\) resulted in -8. Hence, these values do not satisfy the equation. This step is essential as it ensures that the solutions we deem correct truly solve the given equation.

Substitution Method

The substitution method is a fundamental technique in algebra for solving and verifying solutions of equations. It involves replacing a variable with a given value to test if it satisfies the equation.In the problem example, we dealt with the equation \( x^2 - 4x - 5 = 0 \).
Here is how to use the substitution method:

• Choose a value to check, such as \(x = -5\).
• Replace \(x\) in the equation with this value: \((-5)^2 - 4(-5) - 5 = 0\).
• Simplify each component: 25 + 20 - 5.
• Check if these add up to zero. Since they do not (it adds up to 40), \(x = -5\) is not a solution.

The same process applies for the other value, \(x = 1\). Each step involves straightforward arithmetic, which allows us to clearly see whether our values satisfy the equation, making this method both methodical and accurate.

Algebraic Equations

Algebraic equations, like our quadratic equation \( x^2 - 4x - 5 = 0 \), are mathematical statements that equal two expressions.These statements include variables (e.g., \(x\)) and constants (e.g., -4, -5). The primary goal in working with algebraic equations is to find values for the variables that make the equation true.
To understand and solve these equations effectively:

• Recognize the type of equation you're working with—in this case, a quadratic equation, which typically has the form \( ax^2 + bx + c = 0 \).
• Apply mathematical operations to manipulate and simplify the equation to test solutions.
• Use methods like the quadratic formula or factoring to find potential solutions, or test specific values for verification.

Grasping the basics of algebraic equations allows students to tackle more complex problems confidently and develop a strong foundation for further studies in mathematics.