To solve the quadratic equation [tex]\(x^2 - 6x = -58\)[/tex], we first need to set it to the standard form of a quadratic equation, which is [tex]\(ax^2 + bx + c = 0\)[/tex].
The given equation is:
[tex]\[ x^2 - 6x = -58 \][/tex]
Rewriting to standard form:
[tex]\[ x^2 - 6x + 58 = 0 \][/tex]
Now, identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = -6 \][/tex]
[tex]\[ c = 58 \][/tex]
Next, calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-6)^2 - 4(1)(58) \][/tex]
[tex]\[ \Delta = 36 - 232 \][/tex]
[tex]\[ \Delta = -196 \][/tex]
Since the discriminant is negative ([tex]\(\Delta < 0\)[/tex]), the solutions will be complex.
The solutions for a quadratic equation with a negative discriminant are found using the formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Given that [tex]\(\Delta = -196\)[/tex]:
[tex]\[ \sqrt{\Delta} = \sqrt{-196} = 14i \][/tex]
So the solutions are:
[tex]\[ x_1 = \frac{-(-6) + 14i}{2 \cdot 1} = \frac{6 + 14i}{2} = 3 + 7i \][/tex]
[tex]\[ x_2 = \frac{-(-6) - 14i}{2 \cdot 1} = \frac{6 - 14i}{2} = 3 - 7i \][/tex]
Therefore, the solutions of the quadratic equation [tex]\(x^2 - 6x + 58 = 0\)[/tex] are:
[tex]\[ x_1 = 3 + 7i \][/tex]
[tex]\[ x_2 = 3 - 7i \][/tex]
