91Ó°ÊÓ

When solving quadratic equations, the first step is often to isolate the variable term. Let's break down what that means. The goal here is to get the variable (in this case, \(x^2\)) by itself on one side of the equation. For our specific equation, \(\frac{4}{3} x^{2}=48\), we need to remove the fraction \(\frac{4}{3}\) that's multiplying \(x^2\).

To do this, we divide both sides of the equation by \(\frac{4}{3}\). Remember, whatever you do to one side of the equation, you must also do to the other side. This keeps the equation balanced.

So, we divide both sides by \(\frac{4}{3}\):
\[ x^{2} = 48 \div \frac{4}{3} \]
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{4}{3}\) is \(\frac{3}{4}\), so the equation becomes:
\[ x^{2} = 48 \times \frac{3}{4} \]
This step clears the fraction and isolates \(x^2\) on one side.

Once we've isolated \(x^2\), our next step is to simplify the right side of the equation. The simplified form will help us solve for \(x\) more easily. In our example, we have:
\[ x^{2} = 48 \times \frac{3}{4} \]
We multiply 48 by \(\frac{3}{4}\):

• First, multiply 48 by 3: \(48 \times 3 = 144\)
• Next, divide 144 by 4: \(144 \div 4 = 36\)

Now, our equation is simplified to:
\[ x^{2} = 36 \]
Simplifying the algebraic expression clears up the equation, making the next steps more straightforward.

To solve for \(x\), we need to undo the square. This is done by taking the square root of both sides of the equation. Starting from:
\[ x^{2} = 36 \]
We take the square root of both sides:
\[ x = \pm\sqrt{36} \]
The square root of 36 is 6, but don't forget that squaring a positive or a negative number gives the same result. Therefore, we have two solutions:
\[ x = \pm 6 \]

• One solution is \(x = 6\)
• The other is \(x = -6\)

Taking the square root is a crucial step in solving quadratic equations, as it allows us to find both possible values for \(x\). Always remember to include the positive and negative roots when solving equations of this type.