Duplicate Question
The question on this page has been marked as a duplicate question.
Original Question
Use the quadratic formula to solve the quadratic equation 10x2+5x=5x−8 10 𝑥 2 + 5 𝑥 = 5 𝑥 − 8 . Express its solutions in the fo...Question
Use the quadratic formula to solve the quadratic equation 10x2+5x=5x−8
. Express its solutions in the form a±bi
. Express its solutions in the form a±bi
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To solve the quadratic equation \(10x^2 + 5x = 5x - 8\), we first need to rearrange it into standard form. Starting from the given equation:
\[
10x^2 + 5x - 5x + 8 = 0
\]
Simplifying this gives:
\[
10x^2 + 8 = 0
\]
Next, we'll move 8 to the other side:
\[
10x^2 = -8
\]
Now, divide both sides by 10:
\[
x^2 = -\frac{8}{10}
\]
\[
x^2 = -\frac{4}{5}
\]
Now, we can take the square root of both sides. However, since the right side is negative, we will introduce \(i\) (the imaginary unit):
\[
x = \pm \sqrt{-\frac{4}{5}} = \pm \sqrt{\frac{4}{5}} \cdot i = \pm \frac{2}{\sqrt{5}} i
\]
To express the solution in a standard form \(a \pm bi\), we can rationalize the denominator:
\[
x = \pm \frac{2\sqrt{5}}{5} i
\]
Thus, the solutions of the equation are:
\[
x = 0 \pm \frac{2\sqrt{5}}{5} i
\]
Putting it into the form \(a \pm bi\):
\[
x = 0 \pm \frac{2\sqrt{5}}{5} i
\]
Therefore, the solutions in the form \(a \pm bi\) are:
\[
x = 0 + \frac{2\sqrt{5}}{5} i \quad \text{and} \quad x = 0 - \frac{2\sqrt{5}}{5} i
\]
\[
10x^2 + 5x - 5x + 8 = 0
\]
Simplifying this gives:
\[
10x^2 + 8 = 0
\]
Next, we'll move 8 to the other side:
\[
10x^2 = -8
\]
Now, divide both sides by 10:
\[
x^2 = -\frac{8}{10}
\]
\[
x^2 = -\frac{4}{5}
\]
Now, we can take the square root of both sides. However, since the right side is negative, we will introduce \(i\) (the imaginary unit):
\[
x = \pm \sqrt{-\frac{4}{5}} = \pm \sqrt{\frac{4}{5}} \cdot i = \pm \frac{2}{\sqrt{5}} i
\]
To express the solution in a standard form \(a \pm bi\), we can rationalize the denominator:
\[
x = \pm \frac{2\sqrt{5}}{5} i
\]
Thus, the solutions of the equation are:
\[
x = 0 \pm \frac{2\sqrt{5}}{5} i
\]
Putting it into the form \(a \pm bi\):
\[
x = 0 \pm \frac{2\sqrt{5}}{5} i
\]
Therefore, the solutions in the form \(a \pm bi\) are:
\[
x = 0 + \frac{2\sqrt{5}}{5} i \quad \text{and} \quad x = 0 - \frac{2\sqrt{5}}{5} i
\]
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.