Question

Use the quadratic formula to solve the quadratic equation 10x2+5x=5x−8
. Express its solutions in the form a±bi
.(1 point)

Answer 1

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

Starting with the original equation:

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

Subtract \( 5x \) from both sides:

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

This simplifies to:

\[
10x^2 = -8
\]

Now, add \( 8 \) to both sides to set the equation to zero:

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

Next, we can rewrite this as:

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

Here, we have \( a = 10 \), \( b = 0 \), and \( c = 8 \).

Now we can apply the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting in the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-0 \pm \sqrt{0^2 - 4 \cdot 10 \cdot 8}}{2 \cdot 10}
\]

Calculating under the square root:

\[
x = \frac{0 \pm \sqrt{0 - 320}}{20}
\]

This simplifies to:

\[
x = \frac{\pm \sqrt{-320}}{20}
\]

Noting that \( \sqrt{-320} = \sqrt{320}i \):

\[
\sqrt{320} = \sqrt{64 \cdot 5} = 8\sqrt{5}
\]

Thus, we have:

\[
x = \frac{\pm 8\sqrt{5}i}{20}
\]

This can be simplified:

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

Finally, we can write the solutions in the form \( a \pm bi \):

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

Therefore, the solutions to the quadratic equation are:

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