Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=144$$

Step-by-Step Solution

Answer

The solutions are \( x = 12 \) and \( x = -12 \).

Step 1: Understand the Equation

The equation given is a simple quadratic equation: \[ x^{2} = 144 \] We need to find the values of \( x \) that satisfy this equation.

Step 2: Isolate the Unknown

To solve the equation \( x^{2} = 144 \), take the square root of both sides to isolate \( x \). This gives:\[ x = \pm\sqrt{144} \].

Step 3: Simplify the Expression

Calculate the square root of 144. Since 144 is a perfect square, its square root is 12.Thus, \( x = \pm12 \).This means the two solutions are \( x = 12 \) and \( x = -12 \).

Step 4: Graphical Representation

To support the solution graphically, consider the function \( y = x^{2} \) and \( y = 144 \). Plot both equations on the same graph.The intersections of the parabolic curve \( y = x^2 \) with the horizontal line \( y = 144 \) occur at the points \( x = 12 \) and \( x = -12 \).

Key Concepts

Solving Equations

When approaching quadratic equations like \( x^2 = 144 \), the goal is to find the values for "\( x \)" that satisfy the equation. Quadratic equations, which often follow the form \( ax^2 + bx + c = 0 \), can frequently be tackled by moving everything to one side to form a zero on the other. However, in cases like \( x^2 = 144 \), the equation is already in a simple form, allowing us to directly move to solving it.
To isolate the variable \( x \), take the square root of both sides of the equation. This operation simplifies \( x^2 = 144 \) to \( x = \pm \sqrt{144} \). Remember, taking the square root gives us two potential solutions: a positive and a negative root. Therefore, the solutions to the equation \( x^2 = 144 \) are \( x = 12 \) and \( x = -12 \).
This process is fundamental in solving quadratic equations, emphasizing the importance of isolating the variable while maintaining balance between both sides of the equation.

Graphical Representation

Visualizing solutions to quadratic equations can enhance understanding, especially when confirming results. The equation \( x^2 = 144 \) can be graphically represented as two separate functions: \( y = x^2 \) and \( y = 144 \). By plotting these, you showcase how the solutions align with the points of intersection between these curves.

• The function \( y = x^2 \) forms a parabolic curve, bending upwards with the vertex at the origin \((0,0)\).
• Meanwhile, \( y = 144 \) is a horizontal line intersecting the y-axis at \( 144 \).

These graphical intersections occur where the two functions have the same \( y \)-values, that is, where \( x = 12 \) and \( x = -12 \). Hence, the graphical representation not only verifies our algebraic solutions but provides a clear picture of their equivalence.

Square Roots

Square roots serve as a key tool in solving quadratic equations such as \( x^2 = 144 \). To understand this, recall that a square root reverses the squaring process. Taking the square root of a number "undoes" its squared nature, bringing us back to the original number.
When we encounter \( x^2 = 144 \), our goal is to determine which numbers, when squared, produce 144. By applying the square root operation, we find that both \( 12 \) and \( -12 \) are solutions, reflecting the fact that both \( 12^2 \) and \((-12)^2\) equal 144.
Square roots are expressed as \( \sqrt{n} \), and when discussing real numbers, they produce both positive and negative results as long as the original number inside the square root is positive, by convention. This dual nature is vital, ensuring we don't overlook potential solutions when tackling these problems.