To find the values of [tex]\( x \)[/tex] that are the roots of the given equation [tex]\( 3x - 5 = -2x^2 \)[/tex], we follow these steps:
1. Rearrange the equation into standard form:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Rearranging the given equation,
[tex]\[
-2x^2 + 3x - 5 = 0
\][/tex]
or
[tex]\[
2x^2 - 3x + 5 = 0
\][/tex]
by multiplying both sides by -1.
2. Identify the coefficients:
In the quadratic equation [tex]\( 2x^2 - 3x + 5 = 0 \)[/tex], the coefficients are:
[tex]\[
a = 2, \quad b = -3, \quad c = -5
\][/tex]
3. Use the quadratic formula:
The roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting [tex]\( a = 2 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -5 \)[/tex] into the formula:
[tex]\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{3 \pm \sqrt{9 + 40}}{4}
\][/tex]
[tex]\[
x = \frac{3 \pm \sqrt{49}}{4}
\][/tex]
[tex]\[
x = \frac{3 \pm 7}{4}
\][/tex]
4. Calculate the two roots:
- For the positive square root:
[tex]\[
x = \frac{3 + 7}{4} = \frac{10}{4} = 2.5
\][/tex]
- For the negative square root:
[tex]\[
x = \frac{3 - 7}{4} = \frac{-4}{4} = -1
\][/tex]
5. Compare with the given options:
[tex]\[
\text{A. } x = -\frac{5}{2} \quad (\text{equivalent to } -2.5)
\][/tex]
[tex]\[
\text{B. } x = -\frac{1}{3}
\][/tex]
[tex]\[
\text{C. } x = 2
\][/tex]
[tex]\[
\text{D. } x = 1
\][/tex]
We are looking for the roots [tex]\(2.5\)[/tex] and [tex]\(-1\)[/tex] which match with the choices [tex]\(A: x = -2.5\)[/tex] and none of the given options (the second root [tex]\(-1\)[/tex] matches none).
So, the correct answer is:
A. [tex]\( x = -\frac{5}{2} \)[/tex]
There is no direct match for the second root [tex]\( -1 \)[/tex] among the given options. It is possible the question has either a typo in options or assumptions about the chosen [tex]\(x\)[/tex] values. Therefore, only one correct selection based on the given options is A.
