Question

what is the positive and negative solution to -x^2-1+11x=4-3x^2+2x

Answer 1

To solve the equation \(-x^2 - 1 + 11x = 4 - 3x^2 + 2x\), we first want to rearrange it into standard form.

Start by moving all terms to one side of the equation:

\[
-x^2 - 1 + 11x - 4 + 3x^2 - 2x = 0
\]

Now combine like terms:

\[
(-x^2 + 3x^2) + (11x - 2x) + (-1 - 4) = 0
\]

This simplifies to:

\[
2x^2 + 9x - 5 = 0
\]

Next, we can apply the quadratic formula to find the solutions:

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

Here, \(a = 2\), \(b = 9\), and \(c = -5\).

Calculating the discriminant:

\[
b^2 - 4ac = 9^2 - 4(2)(-5) = 81 + 40 = 121
\]

Now plug in the values into the quadratic formula:

\[
x = \frac{{-9 \pm \sqrt{121}}}{2(2)} = \frac{{-9 \pm 11}}{4}
\]

This gives us two solutions:

1. **Positive solution**:
\[
x = \frac{{-9 + 11}}{4} = \frac{2}{4} = \frac{1}{2}
\]

2. **Negative solution**:
\[
x = \frac{{-9 - 11}}{4} = \frac{{-20}}{4} = -5
\]

So the positive solution is \( \frac{1}{2} \) and the negative solution is \(-5\).