To find the possible values of [tex]\(x\)[/tex] given the ratio
[tex]\[
\frac{x^2}{3x + 14} = \frac{1}{2},
\][/tex]
we will solve this equation step-by-step.
1. Set up the equation:
[tex]\[
\frac{x^2}{3x + 14} = \frac{1}{2}.
\][/tex]
2. Clear the fraction by cross-multiplying:
[tex]\[
2x^2 = 3x + 14.
\][/tex]
3. Rearrange the equation to set it to zero:
[tex]\[
2x^2 - 3x - 14 = 0.
\][/tex]
4. Solve this quadratic equation using the quadratic formula:
The quadratic formula is given by
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\][/tex]
where [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -14\)[/tex].
5. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-14) = 9 + 112 = 121.
\][/tex]
6. Find the roots using the quadratic formula:
[tex]\[
x = \frac{-(-3) \pm \sqrt{121}}{2 \cdot 2} = \frac{3 \pm 11}{4}.
\][/tex]
There are two solutions:
[tex]\[
x_1 = \frac{3 + 11}{4} = \frac{14}{4} = 3.5,
\][/tex]
[tex]\[
x_2 = \frac{3 - 11}{4} = \frac{-8}{4} = -2.
\][/tex]
The possible values of [tex]\(x\)[/tex] are:
[tex]\[
x = -2, 3.5.
\][/tex]
