Problem 10 - Elementary and Intermediate Algebra

Question

Check to determine whether the given number is a solution of the given quadratic equation. a. \(x^{2}-4 x=0 ; 4\) b. \(x^{2}-2 x-7=0 ;-2\)

Step-by-Step Solution

Answer

(a) 4 is a solution; (b) -2 is not a solution.

Step 1: Understanding the Problem

We need to check if the given numbers are solutions to their corresponding quadratic equations. This involves substituting the given number into the equation and checking if the equation holds true.

Step 1: Substitute for Part (a)

For the equation \(x^2 - 4x = 0\), check if \(x = 4\) is a solution by substituting \(4\) into the equation. Substitute: \((4)^2 - 4(4) = 0\).

Step 2: Evaluate for Part (a)

Calculate \((4)^2 - 4(4) = 16 - 16 = 0\). Since the left-hand side equals the right-hand side 0, \(x = 4\) is a solution.

Step 3: Substitute for Part (b)

For the equation \(x^2 - 2x - 7 = 0\), check if \(x = -2\) is a solution by substituting \(-2\) into the equation. Substitute: \((-2)^2 - 2(-2) - 7 = 0\).

Step 4: Evaluate for Part (b)

Calculate \((-2)^2 - 2(-2) - 7 = 4 + 4 - 7 = 1\). Since the left-hand side equals 1, which is not 0, \(x = -2\) is not a solution.

Key Concepts

Solution Verification

When you come across a quadratic equation and want to verify if a given number is its solution, you perform something known as "solution verification". This process involves checking whether substituting the number into the equation satisfies it completely. For quadratic equations of the form \(ax^2 + bx + c = 0\), plug the given number into \(x\) and simplify.
Let's briefly go over the steps you might follow:

• Substitute: Replace the variable \(x\) with the given number.
• Calculate: Carry out any arithmetic required to get a value.
• Compare: Check if the result equals zero (since all terms are moved to one side of the equation).

If the equation balances, meaning the left-hand side equals the right-hand side, then the number is indeed a solution. If not, then it isn't a solution. Our exercise perfectly showcases this with two quadratic equations, as shown in the original task.

Substitution Method

The "substitution method" is quite straightforward, especially when dealing with quadratic equations. It's basically plugging in a given number where \(x\) appears in the equation. Let's break it down to understand its simplicity.
Consider this example: If you need to find out whether \(x = 4\) is a solution for the quadratic equation \(x^2 - 4x = 0\), here's how you would use substitution:

• Step 1: Write down the quadratic equation.
• Step 2: Wherever you see \(x\), substitute it with 4.
• Step 3: Perform the math: \( (4)^2 - 4(4)\).

After substituting and simplifying, if the left side of the equation equals zero, then \(x = 4\) is a solution. This method is not only simple but also very effective for verifying solutions to equations. You substitute, simplify, and check - that's all there is to it!

Checking Solutions

Once you've substituted a number into a quadratic equation, the key task remaining is "checking solutions". This step ensures that everything is simplified correctly and that the left-hand side of the equation aligns with the expected result, which is typically zero. Here’s a simple way to check:

• Perform Calculations: After substitution, do the math as fully as you can. Make sure each part of the substituted expression is simplified.
• Review the Results: With the equation reduced, compare the resulting expression with the expected value, which is often zero.
• Draw Conclusions: If the substitution result matches the expected result, you've confirmed the solution. If not, consider the number as not a solution.

In our example exercise, checking showed that \(x = 4\) satisfied the equation \(x^2 - 4x = 0\) because both sides of the equation equaled zero after substitution and simplification. However, \(x = -2\) didn’t solve \(x^2 - 2x - 7 = 0\), showing the importance of accurate checking in verifying whether a number truly is an equation's solution.