Problem 36 - Algebra 1: Concepts and Skills

Question

Solve the equation algebraically. Check your solutions by graphing. $$4 x^{2}=16$$

Step-by-Step Solution

Answer

The solutions to the equation \(4x^{2}=16\) are \(x=2\) and \(x=-2\). The solutions are verified by graphing.

Step 1: Simplify the Equation

Rearrange the equation in order to simplify. This can be done by dividing both sides of the equation by 4 to isolate \(x^2\). The equation now reads \(x^{2}=4\).

Step 2: Solve for x

By taking the square root of both sides of the equation, you can solve for x. Do not forget that a square root has two possible values, positive and negative! Hence, the solutions would be \(x=\sqrt{4}\) and \(x=-\sqrt{4}\), or simplified, \(x=2\) and \(x=-2\).

Step 3: Verify the Solutions by Graphing

Plot the function \(y=4x^{2}-16\). According to the solutions gathered in the step above (\(x=2\) and \(x=-2\)), there should be points where the graph intersects the x-axis (where \(y=0\)). Points at \(x=2\) and \(x=-2\) confirm that the algebraic solutions are correct.

Key Concepts

Solving Quadratics

One of the most common types of equations you'll encounter in algebra are quadratic equations. These take the form of \(ax^2 + bx + c = 0\). Solving quadratics is an important skill which can often involve a few different methods to find the roots or solutions of the equation.
In the original exercise, the quadratic equation is \(4x^2 = 16\). Our first step in solving is simplifying the equation by isolating \(x^2\). This is achieved by dividing both sides by 4, giving us \(x^2 = 4\).
Next, to solve for \(x\), we take the square root of both sides of the equation. It's crucial to remember that when you take the square root, you must consider both the positive and negative roots. Therefore, the solutions for this equation are \(x = 2\) and \(x = -2\). These solutions are the values of \(x\) that satisfy the quadratic equation.

Graphing Solutions

Graphing is a powerful tool to visually verify the solutions of a quadratic equation. When we solve a quadratic equation like \(4x^2 = 16\) using algebraic methods, graphing helps confirm our results.
To graphically check our solutions, we rewrite the equation in a format suitable for graphing: \(y = 4x^2 - 16\). Here, the equation represents a parabola.
When graphed, the places where the parabola intersects the x-axis are the solutions to the quadratic equation. This is because these intersection points indicate where \(y = 0\). In our equation, the graph intersects the x-axis at \(x = 2\) and \(x = -2\), confirming our algebraic findings. Graphing thus not only visualizes the solutions but also provides an additional layer of confirmation.

Square Roots

The square root is a fundamental concept in mathematics, essential for solving quadratic equations. When a number is squared, it is multiplied by itself. The square root operation is the reverse process, used to find the original number that was squared.
In our quadratic equation, once simplified to \(x^2 = 4\), we use square roots to solve for \(x\). The square root of 4 is 2, yet we must consider both \(\sqrt{4}\) and \(-\sqrt{4}\), hence the solutions \(x = 2\) and \(x = -2\).
Including both the positive and negative roots is crucial because they represent all possible solutions to the equation. This principle ensures that no potential solutions are missed during calculations.