Question

a) Determine the root(s) of the equation \(\sqrt{x+7}-4=0\) algebraically. b) Determine the \(x\) -intercept(s) of the graph of the function \(y=\sqrt{x+7}-4\) graphically. c) Explain the connection between the root(s) of the equation and the \(x\) -intercept(s) of the graph of the function.

Short answer

The root and the x-intercept are both at \(x = 9\).

Step-by-step solution

- Isolate the square root term

Start by isolating the square root term on one side of the equation: \(\sqrt{x+7} - 4 = 0\). Add 4 to both sides to get: \(\sqrt{x+7} = 4\).

- Square both sides

To eliminate the square root, square both sides of the equation: \((\sqrt{x+7})^2 = 4^2\). This simplifies to: \(x + 7 = 16\).

- Solve for x

Solve for \(x\) by subtracting 7 from both sides: \(x = 16 - 7\). This gives: \(x = 9\).

Graphical Solution for x-intercept(s)

To find the \(x\)-intercept(s) of the function \(y = \sqrt{x+7} - 4\) graphically, set \(y = 0\) and solve for \(x\) in the same manner as above. This also leads to \(x + 7 = 16\) and results in \(x = 9\).

Connection between root(s) and x-intercept(s)

The root of the equation \(\sqrt{x+7} - 4 = 0\) is the value of \(x\) that makes the function \(y = \sqrt{x+7} - 4\) equal to zero. In other words, it is the \(x\)-intercept of the graph of the function. Both the algebraic and graphical solutions lead to \(x = 9\).

Key concepts

Algebraic Manipulation

Algebraic manipulation is a core skill in solving equations. It involves rearranging equations to isolate the variable you're solving for.
In the given exercise, the goal is to find the value of \(x\) that satisfies the equation \(\sqrt{x+7}-4=0\).
Here’s a clear step-by-step process to isolate \(x\):

• First, add 4 to both sides of the equation: \(\sqrt{x+7} - 4 + 4 = 0 + 4\), which simplifies to \(\sqrt{x+7} = 4\).
• Next, to eliminate the square root, square both sides: \( (\sqrt{x+7})^2 = 4^2 \), resulting in \((x+7) = 16\).
• Finally, solve for \x\ by subtracting 7 from both sides: \( x + 7 - 7 = 16 - 7 \), giving you \( x = 9 \).

By following these algebraic steps, we can successfully isolate and solve for \(x\). Practice these steps often, as they are commonly used in various types of equations.

Square Root Functions

Understanding square root functions is crucial for solving equations that involve square roots.
The function \( y = \sqrt{x+7} - 4\) in the exercise is a great example of a square root function. Key characteristics of square root functions include:

• They only produce non-negative outputs. This means that \( y \) is always greater than or equal to zero.
• The domain of a square root function is restricted to values that make the expression inside the square root non-negative. For \( y = \sqrt{x+7} - 4 \), the domain is \( x \geq -7 \).

When we set \( y = 0 \) to find the \( x\)-intercept, we are essentially solving for the point where the function intersects the x-axis. Thus, solving \( \sqrt{x+7} - 4 = 0 \) gives us the intercept directly.

x-Intercepts

The x-intercepts of a function are the points where the graph crosses the x-axis. These points have coordinates \( (x, 0) \).
For the function \( y = \sqrt{x+7} - 4 \), finding the x-intercepts involves setting \( y = 0 \). This simplifies to solving \( \sqrt{x+7} - 4 = 0 \), the same equation used in the algebraic solution.
Here's how to think about it graphically:

• The x-intercept is found at the point where the graph touches the x-axis, which happens when the output (\( y \)) is zero.
• This point corresponds to the root of the equation, which we already found to be \( x = 9 \).

This shows that the roots of an equation and the x-intercepts of the corresponding function's graph are deeply connected. Both tell you where the function’s output is zero, helping you understand the behavior of the function.