Question
Complete the square to solve 2x^2 + 20x = 10
Answers
GPT-4o mini
To solve the equation \( 2x^2 + 20x = 10 \) by completing the square, follow these steps:
1. **Rearrange the equation:** Start by moving the constant term to the other side of the equation:
\[
2x^2 + 20x - 10 = 0
\]
2. **Simplify by dividing by 2:** This will make the calculations easier:
\[
x^2 + 10x - 5 = 0
\]
3. **Isolate the quadratic and linear terms:** Move the constant term to the right side:
\[
x^2 + 10x = 5
\]
4. **Complete the square:** To complete the square, take half of the coefficient of \( x \) (which is 10), square it, and add it to both sides. Half of 10 is 5, and squaring it gives 25:
\[
x^2 + 10x + 25 = 5 + 25
\]
This simplifies to:
\[
(x + 5)^2 = 30
\]
5. **Take the square root of both sides:**
\[
x + 5 = \pm \sqrt{30}
\]
6. **Solve for \( x \):** Isolate \( x \) by subtracting 5 from both sides:
\[
x = -5 \pm \sqrt{30}
\]
Thus, the solutions to the equation \( 2x^2 + 20x = 10 \) are:
\[
x = -5 + \sqrt{30} \quad \text{and} \quad x = -5 - \sqrt{30}
\]
1. **Rearrange the equation:** Start by moving the constant term to the other side of the equation:
\[
2x^2 + 20x - 10 = 0
\]
2. **Simplify by dividing by 2:** This will make the calculations easier:
\[
x^2 + 10x - 5 = 0
\]
3. **Isolate the quadratic and linear terms:** Move the constant term to the right side:
\[
x^2 + 10x = 5
\]
4. **Complete the square:** To complete the square, take half of the coefficient of \( x \) (which is 10), square it, and add it to both sides. Half of 10 is 5, and squaring it gives 25:
\[
x^2 + 10x + 25 = 5 + 25
\]
This simplifies to:
\[
(x + 5)^2 = 30
\]
5. **Take the square root of both sides:**
\[
x + 5 = \pm \sqrt{30}
\]
6. **Solve for \( x \):** Isolate \( x \) by subtracting 5 from both sides:
\[
x = -5 \pm \sqrt{30}
\]
Thus, the solutions to the equation \( 2x^2 + 20x = 10 \) are:
\[
x = -5 + \sqrt{30} \quad \text{and} \quad x = -5 - \sqrt{30}
\]