Certainly! Let's solve the quadratic equation step by step:
1. Rewrite the equation into standard quadratic form:
Original equation:
[tex]\[
b^2 - 4b - 14 = -2
\][/tex]
Add 2 to both sides to get:
[tex]\[
b^2 - 4b - 14 + 2 = 0
\][/tex]
Simplify:
[tex]\[
b^2 - 4b - 12 = 0
\][/tex]
This is now in the standard quadratic form [tex]\(Ax^2 + Bx + C = 0\)[/tex] where [tex]\(A = 1\)[/tex], [tex]\(B = -4\)[/tex], and [tex]\(C = -12\)[/tex].
2. Identify the coefficients:
[tex]\[
A = 1, \quad B = -4, \quad C = -12
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = B^2 - 4AC
\][/tex]
Substitute the values:
[tex]\[
\Delta = (-4)^2 - 4 \cdot 1 \cdot (-12)
\][/tex]
Simplify:
[tex]\[
\Delta = 16 + 48 = 64
\][/tex]
4. Find the roots using the quadratic formula:
The roots of the quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex] are given by:
[tex]\[
x = \frac{-B \pm \sqrt{\Delta}}{2A}
\][/tex]
Substitute [tex]\(A\)[/tex], [tex]\(B\)[/tex], and the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[
x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 1}
\][/tex]
Simplify:
[tex]\[
x = \frac{4 \pm 8}{2}
\][/tex]
5. Calculate the two possible roots:
First root:
[tex]\[
x_1 = \frac{4 + 8}{2} = \frac{12}{2} = 6
\][/tex]
Second root:
[tex]\[
x_2 = \frac{4 - 8}{2} = \frac{-4}{2} = -2
\][/tex]
6. Summarize the results:
The discriminant is:
[tex]\[
\Delta = 64
\][/tex]
The roots are:
[tex]\[
x_1 = 6 \quad \text{and} \quad x_2 = -2
\][/tex]
Therefore, the solutions to the equation [tex]\(b^2 - 4b - 14 = -2\)[/tex] are [tex]\(b = 6\)[/tex] and [tex]\(b = -2\)[/tex].
