Question

solve 2x^2 +5x+10=0 by completing the square, show your work

Answer 1

To solve the quadratic equation \( 2x^2 + 5x + 10 = 0 \) by completing the square, we can follow these steps:

1. **Divide by the coefficient of \( x^2 \)**: Since the coefficient of \( x^2 \) is 2, we'll divide the entire equation by 2 to simplify it.

\[
x^2 + \frac{5}{2}x + 5 = 0
\]

2. **Isolate the constant term**: Move the constant (5) to the right side of the equation.

\[
x^2 + \frac{5}{2}x = -5
\]

3. **Complete the square**: To complete the square, we need to take half of the coefficient of \( x \) (which is \( \frac{5}{2} \)), square it, and add it to both sides.

Half of \( \frac{5}{2} \) is \( \frac{5}{4} \), and squaring that gives:

\[
\left(\frac{5}{4}\right)^2 = \frac{25}{16}
\]

Now, we will add \( \frac{25}{16} \) to both sides:

\[
x^2 + \frac{5}{2}x + \frac{25}{16} = -5 + \frac{25}{16}
\]

4. **Simplify the right side**: Convert -5 to a fraction with a denominator of 16:

\[
-5 = -\frac{80}{16}
\]

Therefore:

\[
-5 + \frac{25}{16} = -\frac{80}{16} + \frac{25}{16} = -\frac{55}{16}
\]

So our equation becomes:

\[
x^2 + \frac{5}{2}x + \frac{25}{16} = -\frac{55}{16}
\]

5. **Write the left side as a perfect square**:

\[
\left(x + \frac{5}{4}\right)^2 = -\frac{55}{16}
\]

6. **Take the square root of both sides**:

\[
x + \frac{5}{4} = \pm \sqrt{-\frac{55}{16}}
\]

Since we are taking the square root of a negative number, this will involve imaginary numbers:

\[
x + \frac{5}{4} = \pm \frac{\sqrt{55}}{4}i
\]

7. **Solve for \( x \)**: Isolate \( x \):

\[
x = -\frac{5}{4} \pm \frac{\sqrt{55}}{4}i
\]

Thus, the solutions to the equation \( 2x^2 + 5x + 10 = 0 \) are:

\[
x = -\frac{5}{4} + \frac{\sqrt{55}}{4}i \quad \text{and} \quad x = -\frac{5}{4} - \frac{\sqrt{55}}{4}i
\]