To solve the quadratic equation [tex]\(x^2 - 4x = -7\)[/tex] using the quadratic formula, let's first rewrite the equation in standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
1. Rewrite the given equation:
[tex]\[
x^2 - 4x = -7
\][/tex]
Move all terms to one side to get:
[tex]\[
x^2 - 4x + 7 = 0
\][/tex]
Thus, we have:
[tex]\[
a = 1, \quad b = -4, \quad c = 7
\][/tex]
2. Quadratic Formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = (-4)^2 - 4(1)(7) = 16 - 28 = -12
\][/tex]
4. Find the solutions:
Since the discriminant is negative ([tex]\(-12\)[/tex]), the solutions will be complex numbers. Using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-4) \pm \sqrt{-12}}{2 \cdot 1} = \frac{4 \pm \sqrt{-12}}{2} = \frac{4 \pm \sqrt{12i^2}}{2}
\][/tex]
Note that [tex]\(\sqrt{-12} = \sqrt{12}i = 2\sqrt{3}i\)[/tex]. Therefore:
[tex]\[
x = \frac{4 \pm 2\sqrt{3}i}{2}
\][/tex]
Simplifying further:
[tex]\[
x = 2 \pm \sqrt{3}i
\][/tex]
5. Write the final solutions:
[tex]\[
x = 2 + \sqrt{3}i \quad \text{or} \quad x = 2 - \sqrt{3}i
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 - 4x = -7\)[/tex] are:
[tex]\[
x = 2 + i \sqrt{3} \quad \text{or} \quad x = 2 - i \sqrt{3}
\][/tex]
Hence, the correct choice from the given options is:
[tex]\[
x = 2 + i \sqrt{3} \quad \text{or} \quad x = 2 - i \sqrt{3}
\][/tex]
