Problem 25 - Algebra 1

Question

Solve the equation algebraically. Check the solutions graphically. $$ \frac{1}{4} x^{2}=36 $$

Step-by-Step Solution

Answer

The solutions to the equation \(\frac{1}{4} x^2 = 36\) are \(x = 12\) and \(x = -12\) as confirmed by both algebraic solution and graphical representation.

Step 1: Simplify the Equation

First, multiply both sides of the equation \(\frac{1}{4} x^{2} = 36\) by 4 to get rid of the fraction. This simplifies the equation to \(x^{2} = 144\).

Step 2: Solve for x

To solve for \(x\), take the square root of both sides of the equation. Remember that a square root has two possible values -- a positive and a negative value. So the solutions are \(x =\sqrt{144}\) and \(x =-\sqrt{144}\), which simplifies to \(x = 12\) and \(x = -12\).

Step 3: Check the Solutions Graphically

Draw a graph of the equation \(y= \frac{1}{4}x^2 \). The points of intersection of the graph of this equation and the line \(y=36\) are the solutions for the equation \(\frac{1}{4}x^2 = 36\). You should find that the graph intersects the line \(y = 36\) at \(x = 12\) and \(x = -12\), confirming the earlier solutions.

Key Concepts

Algebraic Solution of Equations

Solving quadratic equations algebraically involves manipulating the original equation into a more manageable form. To start, we isolate the variable term on one side to make further operations easier. Take the given equation \( \frac{1}{4} x^{2}=36 \). By multiplying both sides by 4, we get \( x^{2} = 144 \), setting us up to find solutions for \( x \).

Next, we need to consider the properties of squares and square roots. Since squaring a number always gives a non-negative result, taking the square root of a number gives us two possible solutions: one positive and one negative. Apply the square root to both sides of \( x^{2} = 144 \) which yields \( x = 12 \) and \( x = -12 \) as solutions. This method, known as the square root method, is particularly efficient for quadratic equations that can be easily transformed into a perfect square.

Graphical Method in Algebra

The graphical method provides a visual way of finding solutions to equations. For our quadratic equation \( \frac{1}{4} x^{2}=36 \), we graph the corresponding quadratic function \( y= \frac{1}{4}x^2 \). The x-axis represents the variable \( x \) while the y-axis corresponds to the function's value for each \( x \).

The points where the graph of the function intersects with the line \( y = 36 \) are the visual representations of the solutions to the equation. These intersection points occur where the output of the quadratic function equals 36, which can be determined by inspecting the graph. The solutions found through the algebraic method, \( x = 12 \) and \( x = -12 \), should match the intersection points observed on the graph, serving as a confirmation of the correct solutions.

Square Root Method

The square root method is a straightforward approach used when a quadratic equation can be simplified into the form \( x^2 = a \), where \( a \) is a non-negative real number. For our equation, after multiplying through by 4, we arrive at \( x^2 = 144 \).

Applying the square root to both sides, we take into account that \( \sqrt{144} = 12 \) and remember that there must be two solutions: \( x = +\sqrt{144} \) and \( x = -\sqrt{144} \) due to the principle that both positive and negative numbers square to give a positive result. Therefore, our solutions are \( x = 12 \) and \( x = -12 \) respectively. This method bypasses the need for factoring or completing the square, which can save time when dealing with perfect square quadratic equations.

Quadratic Function Graph

The graph of a quadratic function \( y=ax^2+bx+c \) is a parabola. In the given equation \( y = \frac{1}{4}x^2 \), the coefficients \( b \) and \( c \) are zero, resulting in a simple parabolic shape that opens upwards since the coefficient of \( x^2 \) is positive. The vertex of this parabola is at the origin (0,0), and the graph is symmetrical about the y-axis.

When graphed, you can see that the curve gets steeper as the value of \( x \) moves away from zero. Intersection points with horizontal lines, such as \( y=36 \), corresponding to the equation \( \frac{1}{4} x^{2}=36 \) in our exercise, indicate the solutions to the quadratic equation. Notably, the graphical approach not only gives the solutions but also offers insight into the behavior of the quadratic function across different values of \( x \) and \( y \) within a coordinate plane.