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 form a±bi
𝑎
±
𝑏
𝑖
.

Answer 1

To solve the quadratic equation \( 10x^2 + 5x = 5x - 8 \), we first need to rearrange the equation to standard form, \( ax^2 + bx + c = 0 \).

1. Start by subtracting \( 5x \) from both sides:

\[
10x^2 + 5x - 5x = -8
\]

This simplifies to:

\[
10x^2 = -8
\]

2. Next, add \( 8 \) to both sides:

\[
10x^2 + 8 = 0
\]

3. Move \( 8 \) to the right side:

\[
10x^2 = -8
\]

4. Divide each term by \( 10 \) to simplify:

\[
x^2 = -\frac{8}{10} = -\frac{4}{5}
\]

5. Now, we rearrange this to standard form of a quadratic equation:

\[
x^2 + \frac{4}{5} = 0
\]

The equation is:

\[
x^2 + 0x + \frac{4}{5} = 0
\]

Here, \( a = 1 \), \( b = 0 \), and \( c = \frac{4}{5} \).

Now, we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

6. Calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = 0^2 - 4 \cdot 1 \cdot \frac{4}{5} = 0 - \frac{16}{5} = -\frac{16}{5}
\]

7. Now substitute \( a \), \( b \), and the discriminant into the quadratic formula:

\[
x = \frac{-0 \pm \sqrt{-\frac{16}{5}}}{2 \cdot 1}
\]

This simplifies to:

\[
x = \frac{\pm \sqrt{-\frac{16}{5}}}{2}
\]

8. The square root of a negative number can be expressed in terms of \( i \):

\[
\sqrt{-\frac{16}{5}} = \sqrt{16} \cdot \sqrt{-1} / \sqrt{5} = 4i/\sqrt{5}
\]

Substituting back into the equation for \( x \):

\[
x = \frac{\pm \frac{4i}{\sqrt{5}}}{2}
\]

\[
x = \pm \frac{2i}{\sqrt{5}}
\]

9. We can write \( \frac{2i}{\sqrt{5}} \) in a form involving \( a \) and \( b \):

Let \( a = 0 \) and \( b = \frac{2}{\sqrt{5}} \):

Thus, the solutions can be expressed as:

\[
x = 0 \pm \frac{2}{\sqrt{5}} i
\]

In conclusion, the solutions of the quadratic equation \( 10x^2 + 5x = 5x - 8 \) are:

\[
x = 0 \pm \frac{2}{\sqrt{5}} i
\]