Question

7x² - 2 = -65

Explanation:

Step1: Add 2 to both sides

To isolate the term with \(x^2\), we add 2 to both sides of the equation \(7x^2 - 2=-65\).
\[7x^2-2 + 2=-65 + 2\]
Simplifying both sides, we get:
\[7x^2=-63\]

Step2: Divide both sides by 7

To solve for \(x^2\), we divide both sides of the equation \(7x^2 = - 63\) by 7.
\[\frac{7x^2}{7}=\frac{-63}{7}\]
Simplifying both sides, we have:
\[x^2=-9\]

Step3: Analyze the solution

The square of a real number \(x\) is always non - negative (i.e., \(x^2\geq0\) for all real numbers \(x\)). But we have \(x^2=-9\), and \(-9<0\). So, this equation has no real solutions. However, if we consider complex numbers, we can write \(x=\pm\sqrt{-9}=\pm3i\), where \(i=\sqrt{-1}\) is the imaginary unit.

Answer:

If we are considering real numbers, the equation \(7x^{2}-2 = - 65\) has no real solutions. If we are considering complex numbers, the solutions are \(x = 3i\) and \(x=-3i\) (where \(i=\sqrt{-1}\)).