91Ó°ÊÓ

The first step in solving any quadratic equation usually involves isolating the quadratic term. This means you want the term that includes the squared variable on one side of the equation by itself. In the equation given, we need to deal with the term containing the quadratic variable, which is the \(6x^2\).

Start by handling any constants on the same side as the quadratic term. In our exercise: \(6x^2 - 5 = 73\), we see \(-5\) present with \(6x^2\). To get rid of this \(-5\), just add 5 to both sides of the equation.

Doing so, we get:
\[ 6x^2 - 5 + 5 = 73 + 5 \]
This simplifies to:
\[6x^2 = 78 \]

Now, as you can see, the quadratic term \(6x^2\) is all by itself on one side of the equation.

The next step after isolating the quadratic term is to simplify the equation. To simplify means breaking the equation down into its simpler form so it is easier to solve.

From our simplified equation: \[6x^2 = 78\]
We need to isolate \(x^2\). To do this, divide both sides by the coefficient of \(x^2\) (which is 6 in this case).

So, we divide both sides by 6: \[ \frac{6x^2}{6} = \frac{78}{6} \]
When we simplify this, we get:
\[ x^2 = 13 \]
Now we have reduced the quadratic equation to a simpler and more manageable form.

Finally, to solve for \(x\), we need to deal with the squared term. This is done by taking the square root of both sides of the equation.

From our last step:
\[ x^2 = 13 \]
Take the square root of both sides to solve for \(x\):
\[ x = \pm \sqrt{13} \]
Remember, when you take the square root of both sides, you always include both the positive and negative roots. This is because \((\text{positive number})^2\) and \((\text{negative number})^2\) both give the same result.

Hence, the solutions to our original quadratic equation are:
\[ x = \sqrt{13} \ \text{and} \ x = -\text{sqrt{13}} \]
This means that there are two possible values for \(x\) that will satisfy the original equation.