To determine the value of [tex]\(a - b\)[/tex] given the equations [tex]\(a + b = 9\)[/tex] and [tex]\(ab = 14\)[/tex], follow these steps.
#### Step 1: Form a quadratic equation
Given:
[tex]\[ a + b = 9 \][/tex]
[tex]\[ ab = 14 \][/tex]
We can treat [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as the roots of a quadratic equation. The general form of a quadratic equation with roots [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is:
[tex]\[ x^2 - (a + b)x + ab = 0 \][/tex]
Substituting the given values:
[tex]\[ x^2 - 9x + 14 = 0 \][/tex]
#### Step 2: Solve the quadratic equation
We need to find the roots of the quadratic equation [tex]\(x^2 - 9x + 14 = 0\)[/tex]. We can use the quadratic formula:
[tex]\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \][/tex]
For our equation [tex]\(x^2 - 9x + 14 = 0\)[/tex], [tex]\(A = 1\)[/tex], [tex]\(B = -9\)[/tex], and [tex]\(C = 14\)[/tex]. Plugging these values into the quadratic formula:
[tex]\[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(14)}}{2(1)} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{81 - 56}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{25}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm 5}{2} \][/tex]
This results in two solutions:
[tex]\[ x = \frac{9 + 5}{2} = \frac{14}{2} = 7 \][/tex]
[tex]\[ x = \frac{9 - 5}{2} = \frac{4}{2} = 2 \][/tex]
So, the roots [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are 7 and 2.
#### Step 3: Calculate [tex]\(a - b\)[/tex]
Without loss of generality, let [tex]\(a = 7\)[/tex] and [tex]\(b = 2\)[/tex].
[tex]\[ a - b = 7 - 2 = 5 \][/tex]
Thus, the value of [tex]\(a - b\)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
