Question
Solve for x and express the roots in simplest a+bi form:
2+
x
2
5
=
x
6
The roots of the equation in simplest form are □±□i.
2+
x
2
5
=
x
6
The roots of the equation in simplest form are □±□i.
Answers
GPT-4o mini
To solve the equation
\[
2 + \frac{x^2}{5} = \frac{x}{6},
\]
first, we can eliminate the fractions by finding a common denominator for the entire equation. The common denominator of 5 and 6 is 30. Multiply every term of the equation by 30:
\[
30 \cdot 2 + 30 \cdot \frac{x^2}{5} = 30 \cdot \frac{x}{6}.
\]
This simplifies to:
\[
60 + 6x^2 = 5x.
\]
Next, we rearrange the equation into standard quadratic form:
\[
6x^2 - 5x + 60 = 0.
\]
Now, we apply the quadratic formula, given by
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where in our case \(a = 6\), \(b = -5\), and \(c = 60\). First, we calculate the discriminant \(b^2 - 4ac\):
\[
b^2 - 4ac = (-5)^2 - 4(6)(60) = 25 - 1440 = -1415.
\]
Since the discriminant is negative, the roots will be complex. Now we substitute into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{-1415}}{2(6)} = \frac{5 \pm \sqrt{1415}i}{12}.
\]
Thus, simplifying the expression, we have:
\[
x = \frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
Therefore, the roots of the equation in simplest form are:
\[
\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
So the answer can be expressed as:
\[
\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
Thus, the answer is:
\[
\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
Final answer:
The roots of the equation in simplest form are \(\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i\).
\[
2 + \frac{x^2}{5} = \frac{x}{6},
\]
first, we can eliminate the fractions by finding a common denominator for the entire equation. The common denominator of 5 and 6 is 30. Multiply every term of the equation by 30:
\[
30 \cdot 2 + 30 \cdot \frac{x^2}{5} = 30 \cdot \frac{x}{6}.
\]
This simplifies to:
\[
60 + 6x^2 = 5x.
\]
Next, we rearrange the equation into standard quadratic form:
\[
6x^2 - 5x + 60 = 0.
\]
Now, we apply the quadratic formula, given by
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where in our case \(a = 6\), \(b = -5\), and \(c = 60\). First, we calculate the discriminant \(b^2 - 4ac\):
\[
b^2 - 4ac = (-5)^2 - 4(6)(60) = 25 - 1440 = -1415.
\]
Since the discriminant is negative, the roots will be complex. Now we substitute into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{-1415}}{2(6)} = \frac{5 \pm \sqrt{1415}i}{12}.
\]
Thus, simplifying the expression, we have:
\[
x = \frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
Therefore, the roots of the equation in simplest form are:
\[
\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
So the answer can be expressed as:
\[
\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
Thus, the answer is:
\[
\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i.
\]
Final answer:
The roots of the equation in simplest form are \(\frac{5}{12} \pm \frac{\sqrt{1415}}{12} i\).