Problem 88 - Algebra 1

Question

Solve the equation. $$16+x^{2}=64$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 6.93\) and \(x = -6.93\).

Step 1: Isolate the quadratic term

First, isolate the \(x^{2}\) term by subtracting 16 from both sides of the equation. That gives \(x^{2}= 64 - 16 = 48\)

Step 2: Determine the square root

Next, find the square root of 48. The square root of 48 yields two potential solutions, as we can have a positive root and a negative root. Hence, the solution for \(x^{2}=48\) are \(x = \sqrt{48}\) and \(x = -\sqrt{48}\). Given that \(\sqrt{48} \approx 6.93\), the values for x are \(x = 6.93\) and \(x = -6.93\).

Step 3: Check the solutions if necessary

Given that the question does not require checking the solution, this step is not necessary for this particular exercise. However, in general, it is always a good idea to substitute the computed values back into the original equation to verify if they're correct.

Key Concepts

Isolating Terms

When solving quadratic equations, a great first step is isolating the term that contains the variable. In the equation \(16 + x^{2} = 64\), this means getting \(x^{2}\) by itself on one side of the equation. To do this, subtract 16 from both sides. Think of this like moving blocks from one side of the scale to another to keep it balanced. Here’s how it works:

• Original equation: \(16 + x^{2} = 64\)
• Subtract 16 from both sides: \(x^{2} = 64 - 16\)
• Resulting in: \(x^{2} = 48\)

By isolating \(x^{2}\), we simplify the problem, making it easier to find a solution. This step is crucial because it sets us up to tackle the equation with square roots next.

Square Roots

Once you have isolated the variable term, applying the square root is the next step. Since we found \(x^{2} = 48\), we now need to determine what \(x\) is. Taking the square root of both sides gives us the possible values for \(x\). This is because squaring a number and taking the square root are reverse operations, much like how addition and subtraction undo each other.

• Finding square root: \(x = \pm \sqrt{48}\)
• Approximate value: \(\sqrt{48} \approx 6.93\)

This is the turning point where math reveals multiple solutions because every positive number has both a positive and a negative square root. This leads us directly to understanding positive and negative solutions.

Positive and Negative Solutions

Whenever we find a square root, we discover two potential solutions: one positive and one negative. This stems from the fact that both positive and negative numbers squared yield the same result. In the case of \(x = \sqrt{48}\) and \(x = -\sqrt{48}\), we get:

• Positive solution: \(x = 6.93\)
• Negative solution: \(x = -6.93\)

It is vital to remember that you should account for both roots when solving equations involving \(x^{2}\). Ignoring the negative solution can result in incomplete solutions. Both solutions are valid as long as they satisfy the original equation. Practicing checking both solutions will help ensure accuracy in solving quadratic equations.