Let's solve the given quadratic equation step-by-step.
The given equation is:
[tex]\[ x^2 - 7x - 14 = -6 \][/tex]
### Step 1: Convert the equation to standard form
To convert the given equation to standard form, we need to move all the terms to one side of the equation, resulting in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
We start by adding 6 to both sides of the equation:
[tex]\[ x^2 - 7x - 14 + 6 = 0 \][/tex]
[tex]\[ x^2 - 7x - 8 = 0 \][/tex]
Now, the equation is in standard form, where:
[tex]\[ a = 1, \quad b = -7, \quad c = -8 \][/tex]
### Step 2: Apply the quadratic formula
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients from the standard form of the equation. Plugging in the values we have:
[tex]\[ a = 1, \quad b = -7, \quad c = -8 \][/tex]
First, calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant:
[tex]\[ \Delta = (-7)^2 - 4(1)(-8) \][/tex]
[tex]\[ \Delta = 49 + 32 \][/tex]
[tex]\[ \Delta = 81 \][/tex]
Since the discriminant is 81, we proceed to calculate the two solutions using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the discriminant and the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-7) \pm \sqrt{81}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{7 \pm 9}{2} \][/tex]
This will give us two solutions:
[tex]\[ x_1 = \frac{7 + 9}{2} = \frac{16}{2} = 8 \][/tex]
[tex]\[ x_2 = \frac{7 - 9}{2} = \frac{-2}{2} = -1 \][/tex]
### Conclusion
So the solutions to the equation are:
[tex]\[ x = 8 \quad \text{or} \quad x = -1 \][/tex]
Therefore, the final answers are:
[tex]\[ x = 8 \quad \text{or} \quad x = -1 \][/tex]
