Question
3x^2 - 8x +2 =0 solve by corresponding square method
Answers
GPT-4o mini
To solve the quadratic equation \(3x^2 - 8x + 2 = 0\) using the completing the square method, follow these steps:
1. **Divide the entire equation by the coefficient of \(x^2\)** (which is 3 in this case) to make the leading coefficient 1:
\[
x^2 - \frac{8}{3}x + \frac{2}{3} = 0
\]
2. **Rearrange the equation** to isolate the constant on one side:
\[
x^2 - \frac{8}{3}x = -\frac{2}{3}
\]
3. **Complete the square** on the left side. To do this, take half of the coefficient of \(x\) (which is \(-\frac{8}{3}\)), square it, and add it to both sides.
Half of \(-\frac{8}{3}\) is \(-\frac{4}{3}\). Squaring it gives:
\[
\left(-\frac{4}{3}\right)^2 = \frac{16}{9}
\]
Now add \(\frac{16}{9}\) to both sides:
\[
x^2 - \frac{8}{3}x + \frac{16}{9} = -\frac{2}{3} + \frac{16}{9}
\]
To combine the right side, convert \(-\frac{2}{3}\) to a fraction with a denominator of 9:
\[
-\frac{2}{3} = -\frac{6}{9}
\]
Therefore:
\[
-\frac{6}{9} + \frac{16}{9} = \frac{10}{9}
\]
Now we have:
\[
x^2 - \frac{8}{3}x + \frac{16}{9} = \frac{10}{9}
\]
4. **Rewrite the left side as a perfect square**:
\[
\left(x - \frac{4}{3}\right)^2 = \frac{10}{9}
\]
5. **Take the square root of both sides**:
\[
x - \frac{4}{3} = \pm \sqrt{\frac{10}{9}}
\]
This simplifies to:
\[
x - \frac{4}{3} = \pm \frac{\sqrt{10}}{3}
\]
6. **Add \(\frac{4}{3}\) to both sides** to solve for \(x\):
\[
x = \frac{4}{3} \pm \frac{\sqrt{10}}{3}
\]
7. **Simplify** to express the solutions:
\[
x = \frac{4 \pm \sqrt{10}}{3}
\]
Thus, the solutions to the equation \(3x^2 - 8x + 2 = 0\) are:
\[
x = \frac{4 + \sqrt{10}}{3} \quad \text{and} \quad x = \frac{4 - \sqrt{10}}{3}
\]
1. **Divide the entire equation by the coefficient of \(x^2\)** (which is 3 in this case) to make the leading coefficient 1:
\[
x^2 - \frac{8}{3}x + \frac{2}{3} = 0
\]
2. **Rearrange the equation** to isolate the constant on one side:
\[
x^2 - \frac{8}{3}x = -\frac{2}{3}
\]
3. **Complete the square** on the left side. To do this, take half of the coefficient of \(x\) (which is \(-\frac{8}{3}\)), square it, and add it to both sides.
Half of \(-\frac{8}{3}\) is \(-\frac{4}{3}\). Squaring it gives:
\[
\left(-\frac{4}{3}\right)^2 = \frac{16}{9}
\]
Now add \(\frac{16}{9}\) to both sides:
\[
x^2 - \frac{8}{3}x + \frac{16}{9} = -\frac{2}{3} + \frac{16}{9}
\]
To combine the right side, convert \(-\frac{2}{3}\) to a fraction with a denominator of 9:
\[
-\frac{2}{3} = -\frac{6}{9}
\]
Therefore:
\[
-\frac{6}{9} + \frac{16}{9} = \frac{10}{9}
\]
Now we have:
\[
x^2 - \frac{8}{3}x + \frac{16}{9} = \frac{10}{9}
\]
4. **Rewrite the left side as a perfect square**:
\[
\left(x - \frac{4}{3}\right)^2 = \frac{10}{9}
\]
5. **Take the square root of both sides**:
\[
x - \frac{4}{3} = \pm \sqrt{\frac{10}{9}}
\]
This simplifies to:
\[
x - \frac{4}{3} = \pm \frac{\sqrt{10}}{3}
\]
6. **Add \(\frac{4}{3}\) to both sides** to solve for \(x\):
\[
x = \frac{4}{3} \pm \frac{\sqrt{10}}{3}
\]
7. **Simplify** to express the solutions:
\[
x = \frac{4 \pm \sqrt{10}}{3}
\]
Thus, the solutions to the equation \(3x^2 - 8x + 2 = 0\) are:
\[
x = \frac{4 + \sqrt{10}}{3} \quad \text{and} \quad x = \frac{4 - \sqrt{10}}{3}
\]