Question

-5x^2 + 100x = 1400

Answer 1

To solve the equation \(-5x^2 + 100x = 1400\), we can start by rearranging it into standard quadratic form (ax² + bx + c = 0).

First, move 1400 to the left side of the equation:

\[
-5x^2 + 100x - 1400 = 0
\]

Next, we can divide the entire equation by -5 to simplify it:

\[
x^2 - 20x + 280 = 0
\]

Now, we can apply the quadratic formula, which is given by:

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

For our equation \(a = 1\), \(b = -20\), and \(c = 280\).

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

\[
b^2 - 4ac = (-20)^2 - 4(1)(280) = 400 - 1120 = -720
\]

Since the discriminant is negative (\(-720\)), this means there are no real solutions to the equation, only complex solutions.

We can continue to find the complex solutions:

\[
x = \frac{20 \pm \sqrt{-720}}{2}
\]

\[
x = \frac{20 \pm i\sqrt{720}}{2}
\]

Now simplify \(\sqrt{720}\):

\[
\sqrt{720} = \sqrt{144 \cdot 5} = 12\sqrt{5}
\]

Substituting this back into the equation gives us:

\[
x = \frac{20 \pm 12i\sqrt{5}}{2}
\]

\[
x = 10 \pm 6i\sqrt{5}
\]

Thus, the solutions to the equation \(-5x^2 + 100x = 1400\) are:

\[
x = 10 + 6i\sqrt{5} \quad \text{and} \quad x = 10 - 6i\sqrt{5}
\]