To solve the quadratic equation [tex]\(x^2 + 6x - 8 = 0\)[/tex] and find the solutions [tex]\(A\)[/tex] and [tex]\(B\)[/tex] such that [tex]\(A < B\)[/tex], we can follow these steps:
1. Identify the general form of the quadratic equation:
A quadratic equation generally has the form [tex]\(ax^2 + bx + c = 0\)[/tex]. In this case, the given equation is:
[tex]\[
x^2 + 6x - 8 = 0
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = -8\)[/tex].
2. Use the quadratic formula to find the roots:
The quadratic formula, which solves the equation [tex]\(ax^2 + bx + c = 0\)[/tex], is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula yields:
[tex]\[
x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-6 \pm \sqrt{36 + 32}}{2}
\][/tex]
[tex]\[
x = \frac{-6 \pm \sqrt{68}}{2}
\][/tex]
[tex]\[
x = \frac{-6 \pm 2\sqrt{17}}{2}
\][/tex]
3. Simplify the expression for the roots:
[tex]\[
x = -3 \pm \sqrt{17}
\][/tex]
Therefore, the two solutions are:
[tex]\[
x_1 = -3 + \sqrt{17}
\][/tex]
and
[tex]\[
x_2 = -3 - \sqrt{17}
\][/tex]
4. Identify which solution is smaller and which is larger:
We need to determine which solution is [tex]\(A\)[/tex] (the smaller root) and which is [tex]\(B\)[/tex] (the larger root). Clearly:
[tex]\[
-3 - \sqrt{17}
\][/tex]
Thus, the solutions are:
[tex]\[
A = -3 - \sqrt{17}
\][/tex]
[tex]\[
B = -3 + \sqrt{17}
\][/tex]
Therefore, the solutions are:
[tex]\[
A = -\sqrt{17} - 3 \quad \text{(the smaller root)}
\][/tex]
[tex]\[
B = -3 + \sqrt{17} \quad \text{(the larger root)}
\][/tex]
