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Chapter 1: Problem 9

Solve the equation both algebraically and graphically. $$x^{2}-32=0$$

Short Answer

Expert verified
The solutions are \( x = 4\sqrt{2} \) and \( x = -4\sqrt{2} \).

Step by step solution

01

Set the equation to zero

We begin by ensuring that the equation is properly set to zero: \( x^2 - 32 = 0 \). This equation is already set to be solved.
02

Isolate and solve for x

Add 32 to both sides to isolate \( x^2 \): \( x^2 = 32 \). Then take the square root of both sides to solve for \( x \): \( x = \pm \sqrt{32} \). Simplifying gives \( x = \pm 4\sqrt{2} \).
03

Verify the solution algebraically

Substitute \( x = 4\sqrt{2} \) and \( x = -4\sqrt{2} \) back into the original equation to verify: \( (4\sqrt{2})^2 - 32 = 0 \) and \( (-4\sqrt{2})^2 - 32 = 0 \). Both simplify to \( 32 - 32 = 0 \), thus verifying each solution.
04

Solve graphically

Graph the function \( f(x) = x^2 - 32 \). Identify where this graph intersects the x-axis, which represents the solutions. The intersection points are \( x = 4\sqrt{2} \) and \( x = -4\sqrt{2} \).
05

Verify graphical solution

Check if the x-values from the graph's intersection points match the algebraic solutions; they should confirm the solutions \( x = \pm 4\sqrt{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Solution
To solve a quadratic equation algebraically, the goal is to isolate the variable to find its possible values. In this exercise, the given equation is \(x^2 - 32 = 0\). First, we ensure that the equation is set to zero. It's already arranged properly, so we can move on to isolating \(x^2\).
  • Add 32 to both sides: \(x^2 = 32\).
  • Take the square root of both sides to get \(x = \pm \sqrt{32}\).
Simplify the square root to make it easier to manage. The simplification of \(\sqrt{32}\) is \(4\sqrt{2}\), so the solutions for \(x\) are \(x = \pm 4\sqrt{2}\). This method reveals the solutions directly through manipulation of the equation.
Graphical Solution
Solving a quadratic equation graphically can help provide a visual understanding of the solutions. This involves drawing the graph of the function defined by the quadratic equation.For our equation, we graph \(f(x) = x^2 - 32\). The graph of this function is a parabola that opens upwards. To find the solutions graphically, look for the points where the parabola intersects the x-axis. These intersection points represent the x-values where \(f(x) = 0\).
  • The x-intercepts of the graph correspond to the solutions \(x = 4\sqrt{2}\) and \(x = -4\sqrt{2}\).
Using a graph can make the solutions more intuitive, especially if you're visually inclined.
Verifying Solutions
Verification is an important step to ensure that the found solutions satisfy the original equation. This can be done both algebraically and graphically.**Algebraic Verification**To verify the solutions algebraically, substitute \(x = 4\sqrt{2}\) and \(x = -4\sqrt{2}\) back into the original equation \(x^2 - 32 = 0\).
  • Calculate \((4\sqrt{2})^2 = 32\) and substitute back: \(32 - 32 = 0\).
  • For \(-4\sqrt{2}\), the calculation is the same: \((-4\sqrt{2})^2 = 32\) and \(32 - 32 = 0\).
Both calculations confirm that the solutions satisfy the original equation.**Graphical Verification**Graphically, verification involves checking that the x-intercepts of the plotted function are indeed \(x = 4\sqrt{2}\) and \(x = -4\sqrt{2}\). These intercepts exactly match the solutions obtained algebraically, thus verifying their correctness visually.

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Most popular questions from this chapter

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