Answer :

Sure, let's solve the equation step-by-step.

We have the equation:

[tex]\[ 27x^2 = 60 \][/tex]

1. Divide both sides by 27 to isolate [tex]\(x^2\)[/tex]:

[tex]\[ x^2 = \frac{60}{27} \][/tex]

2. Simplify [tex]\(\frac{60}{27}\)[/tex]:

- The greatest common divisor (GCD) of 60 and 27 is 3.
- Divide both the numerator and the denominator by 3:

[tex]\[ x^2 = \frac{60 \div 3}{27 \div 3} = \frac{20}{9} \][/tex]

3. Take the square root of both sides to solve for [tex]\(x\)[/tex]:

[tex]\[ x = \pm \sqrt{\frac{20}{9}} \][/tex]

4. Simplify the square root:

- The square root of [tex]\(\frac{20}{9}\)[/tex] can be split into:

[tex]\[ x = \pm \frac{\sqrt{20}}{\sqrt{9}} \][/tex]

- Since [tex]\(\sqrt{9} = 3\)[/tex], we have:

[tex]\[ x = \pm \frac{\sqrt{20}}{3} \][/tex]

- Simplify [tex]\(\sqrt{20}\)[/tex] as [tex]\( \sqrt{4 \times 5} = 2\sqrt{5} \)[/tex].

5. Plug back into the equation:

[tex]\[ x = \pm \frac{2\sqrt{5}}{3} \][/tex]

This gives us the final solutions:

[tex]\[ x = -\frac{2\sqrt{5}}{3} \quad \text{and} \quad x = \frac{2\sqrt{5}}{3} \][/tex]

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