Question

Solve the radical equation
3
x
2
−
10
5
+
7
=
4
3
5

x
2
−10
​
+7=4 to find the two solutions. Separate your answers with a comma.

Answer 1

To solve the equation

\[
\frac{3x^2 - 10}{5} + 7 = 4,
\]

let's first isolate the fraction on one side.

1. Subtract 7 from both sides:

\[
\frac{3x^2 - 10}{5} = 4 - 7
\]
\[
\frac{3x^2 - 10}{5} = -3.
\]

2. Now, eliminate the fraction by multiplying both sides by 5:

\[
3x^2 - 10 = -15.
\]

3. Next, add 10 to both sides:

\[
3x^2 = -15 + 10
\]
\[
3x^2 = -5.
\]

4. Divide both sides by 3:

\[
x^2 = -\frac{5}{3}.
\]

Since \(x^2\) cannot be negative in real numbers, this equation has no real solutions.

However, if you are looking for complex solutions:

\[
x^2 = -\frac{5}{3},
\]

then taking the square root of both sides gives:

\[
x = \pm i \sqrt{\frac{5}{3}}.
\]

Thus, the two complex solutions in terms of real numbers and imaginary units are:

\[
x = i \sqrt{\frac{5}{3}}, -i \sqrt{\frac{5}{3}}.
\]

If we want to express these more neatly:

\[
x = \frac{\sqrt{15}}{3} i, -\frac{\sqrt{15}}{3} i.
\]

So, the final answers are:

\[
\frac{\sqrt{15}}{3} i, -\frac{\sqrt{15}}{3} i.
\]