Question
2x^2+5x+6=0 by completing the square root
Answers
GPT-4o mini
To solve the quadratic equation \( 2x^2 + 5x + 6 = 0 \) by completing the square, follow these steps:
1. **Divide through by the coefficient of \(x^2\)** (which is 2 in this case) to make the coefficient of \(x^2\) equal to 1:
\[
x^2 + \frac{5}{2}x + 3 = 0
\]
2. **Isolate the constant on one side**:
\[
x^2 + \frac{5}{2}x = -3
\]
3. **Complete the square**:
To complete the square, take half of the coefficient of \(x\) (which is \(\frac{5}{2}\)), square it, and add it to both sides of the equation. Half of \(\frac{5}{2}\) is \(\frac{5}{4}\), and squaring it gives \(\left(\frac{5}{4}\right)^2 = \frac{25}{16}\).
Add \(\frac{25}{16}\) to both sides:
\[
x^2 + \frac{5}{2}x + \frac{25}{16} = -3 + \frac{25}{16}
\]
To simplify the right side, convert -3 to sixteenth parts:
\[
-3 = -\frac{48}{16}
\]
So,
\[
-3 + \frac{25}{16} = -\frac{48}{16} + \frac{25}{16} = -\frac{23}{16}
\]
4. **Rewrite the left side as a square**:
\[
\left(x + \frac{5}{4}\right)^2 = -\frac{23}{16}
\]
5. **Take the square root of both sides**:
\[
x + \frac{5}{4} = \pm \sqrt{-\frac{23}{16}}
\]
This gives:
\[
x + \frac{5}{4} = \pm \frac{\sqrt{23}i}{4}
\]
6. **Solve for \(x\)**:
\[
x = -\frac{5}{4} \pm \frac{\sqrt{23}i}{4}
\]
7. **Combine the results**:
\[
x = -\frac{5 \pm \sqrt{23}i}{4}
\]
Thus, the solutions to the equation \(2x^2 + 5x + 6 = 0\) are:
\[
x = -\frac{5 + \sqrt{23}i}{4} \quad \text{and} \quad x = -\frac{5 - \sqrt{23}i}{4}
\]
1. **Divide through by the coefficient of \(x^2\)** (which is 2 in this case) to make the coefficient of \(x^2\) equal to 1:
\[
x^2 + \frac{5}{2}x + 3 = 0
\]
2. **Isolate the constant on one side**:
\[
x^2 + \frac{5}{2}x = -3
\]
3. **Complete the square**:
To complete the square, take half of the coefficient of \(x\) (which is \(\frac{5}{2}\)), square it, and add it to both sides of the equation. Half of \(\frac{5}{2}\) is \(\frac{5}{4}\), and squaring it gives \(\left(\frac{5}{4}\right)^2 = \frac{25}{16}\).
Add \(\frac{25}{16}\) to both sides:
\[
x^2 + \frac{5}{2}x + \frac{25}{16} = -3 + \frac{25}{16}
\]
To simplify the right side, convert -3 to sixteenth parts:
\[
-3 = -\frac{48}{16}
\]
So,
\[
-3 + \frac{25}{16} = -\frac{48}{16} + \frac{25}{16} = -\frac{23}{16}
\]
4. **Rewrite the left side as a square**:
\[
\left(x + \frac{5}{4}\right)^2 = -\frac{23}{16}
\]
5. **Take the square root of both sides**:
\[
x + \frac{5}{4} = \pm \sqrt{-\frac{23}{16}}
\]
This gives:
\[
x + \frac{5}{4} = \pm \frac{\sqrt{23}i}{4}
\]
6. **Solve for \(x\)**:
\[
x = -\frac{5}{4} \pm \frac{\sqrt{23}i}{4}
\]
7. **Combine the results**:
\[
x = -\frac{5 \pm \sqrt{23}i}{4}
\]
Thus, the solutions to the equation \(2x^2 + 5x + 6 = 0\) are:
\[
x = -\frac{5 + \sqrt{23}i}{4} \quad \text{and} \quad x = -\frac{5 - \sqrt{23}i}{4}
\]