Problem 389 - Elementary Algebra

Question

In the following exercises, check whether the given values are solutions. For the equation \(\sqrt{x+12}=x:\) (a) Is \(x=4 \quad a\) solution? (b) Is \(x=-3\) a solution?

Step-by-Step Solution

Answer

x = 4 is a solution. x = -3 is not a solution.

Step 1 - Substitute x = 4 into the equation

First, substitute the value of x with 4 into the equation. The equation is \ \ \ \ \ \ \( \sqrt{x+12}=x \). So we get: \ \( \sqrt{4+12} = 4 \)

Step 2 - Simplify the equation with x = 4

Next, simplify under the square root: \ \( \sqrt{16} = 4 \). \ Since \( \sqrt{16} = 4 \), this is true. So, \( x = 4 \) is a solution.

Step 3 - Substitute x = -3 into the equation

Now substitute the value of x with -3 into the same equation: \ \( \sqrt{-3+12} = -3 \). This simplifies to: \ \( \sqrt{9} = -3 \)

Step 4 - Simplify the equation with x = -3

Simplify under the square root: \ \( \sqrt{9} = 3 \). \ Since \( 3 eq -3 \), this is false. Therefore, \( x = -3 \) is not a solution.

Key Concepts

equation solving

Equation solving is a fundamental concept in algebra that involves finding values for variables that satisfy given mathematical statements. Understanding how to solve an equation is crucial for progressing in math. When you solve an equation, you are essentially unravelling a puzzle to find the value that makes the equation true.
In the provided exercise, we are given the equation \(\sqrt{x+12}=x\). We need to check if certain values, like 4 or -3, make the equation true.
It's essential to be systematic:

• Start by substituting the provided value into the equation.
• Then, perform any necessary operations to simplify.
• Finally, determine if both sides of the equation are equal.

By following these steps, you can verify if a specific value is indeed a solution to the equation.

substitution method

The substitution method is a powerful technique used to solve equations by replacing variables with given values. This method helps in identifying whether certain values satisfy the equation.
Let's break down the substitution method using our example equation \(\sqrt{x+12}=x\). We need to check two values: 4 and -3.

• First, implement substitution by placing x = 4: \( \sqrt{4+12}=4\), which simplifies to \( \sqrt{16}=4\). It turns out to be true, meaning 4 is a correct solution.
• Next, substitute x with -3: \( \sqrt{-3+12}=-3\), which simplifies to \( \sqrt{9}=-3\). This results in \(3 eq -3\), hence -3 is not a solution.

The substitution method streamlines the process, making equation solving more manageable and precise.

square roots

Square roots are values that, when multiplied by themselves, give the original number. In other words, the square root of a number \( n \) is a value that satisfies the equation \( n = x^2 \).
In our example, we work with the square root function within equations. Understanding this function is critical:

• For instance, \( \sqrt{16} = 4\) because \( 4 \times 4 = 16 \).
• Note the principal square root, which is always non-negative. This is why the solution \( x=-3 \) did not work. The square root of 9 is 3, not -3.
• Square roots are integral in ensuring both sides of the equation balance properly.

By mastering the concept of square roots, one can solve more advanced algebraic equations accurately.