Let's solve the equation step-by-step to find the possible values of [tex]\(x\)[/tex].
Given the ratio:
[tex]\[ \frac{x^2}{3x + 14} = \frac{1}{2} \][/tex]
We start by setting up the equation based on the given ratio:
[tex]\[ \frac{x^2}{3x + 14} = \frac{1}{2} \][/tex]
Now, we clear the fraction by cross multiplying:
[tex]\[ 2x^2 = 3x + 14 \][/tex]
Next, we rearrange the equation to form a standard quadratic equation:
[tex]\[ 2x^2 - 3x - 14 = 0 \][/tex]
Now we solve this quadratic equation using the quadratic formula:
[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].
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = (-3)^2 - 4(2)(-14) \][/tex]
[tex]\[ b^2 - 4ac = 9 + 112 \][/tex]
[tex]\[ b^2 - 4ac = 121 \][/tex]
Next, we take the square root of the discriminant:
[tex]\[ \sqrt{121} = 11 \][/tex]
Now, we apply the quadratic formula:
[tex]\[ x = \frac{-(-3) \pm 11}{2(2)} \][/tex]
[tex]\[ x = \frac{3 \pm 11}{4} \][/tex]
This gives us two possible solutions for [tex]\(x\)[/tex]:
1. [tex]\( x = \frac{3 + 11}{4} = \frac{14}{4} = 3.5 \)[/tex]
2. [tex]\( x = \frac{3 - 11}{4} = \frac{-8}{4} = -2 \)[/tex]
Therefore, the possible values of [tex]\(x\)[/tex] are:
[tex]\[ x = -2 \quad \text{or} \quad x = 3.5 \][/tex]
