Choices C (seven plus I) \( (7+i) \) and D (seven minus I) \( (7-i) \) should be selected.
To find the solution to this quadratic equation, which is given in the form A X squared plus B X plus C = 0, \( ax^2+bx+c=0, \) you can use the quadratic formula:x = fraction with numerator of the opposite of B plus or minus the square root of B squared minus 4 A C and denominator of 2 A
$$ \Large x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} $$
In the problem, A equals 1, B equals negative 14, and c equals 50. \( a=1,b=-14,\text{ and }c=50. \) So substituting those values in the formula and simplifying, first under the square root:X equals fraction with numerator of 14 plus or minus the square root of negative 14 squared minus 4 open parenthesis 1 close parenthesis open parenthesis 50 close parenthesis and denominator of 2 open parenthesis 1 close parenthesis
$$ x=\frac{14\pm \sqrt{(-14)^2-4(1)(50)}}{2(1)} $$
x equals fraction with numerator of 14 plus or minus the square root of 1 hundred 96 minus 200 and denominator of 2 $$ x=\frac{14\pm \sqrt{196-200}}{2} $$
x equals fraction with numerator of 14 plus or minus the square root of negative 4 and denominator of 2 $$ x=\frac{14\pm\sqrt{-4}}{2} $$
x equals fraction with numerator of 14 plus or minus 2 square root of negative 1 and denominator of 2 $$ x=\frac{14\pm 2\sqrt{-1}}{2} $$
x equals fraction with numerator of 14 plus or minus 2 I and denominator of 2 $$ x=\frac{14\pm 2i}{2} $$
Finally, simplifying that gives x equals 7 plus or minus I. \( x=7\pm i. \)
Other approaches to the solution are possible, such as completing the square. The coefficient on the x squared \( x^2 \) term is already 1, so there's no need to divide the equation by the coefficient.x squared minus 14 x plus 50 equals 0.
$$ x^2-14x+50=0. $$
Start by moving the constant term to the other side of the equation, taking half of the coefficient of x, \( x, \) squaring it, and adding it to both sides of the equation.x squared minus 14 x equals negative 50
$$ x^2 - 14x = -50 $$
x squared minus 14 x plus open parenthesis negative 14 halves close parenthesis squared equals negative 50 plus open parenthesis negative 14 halves close parenthesis squared $$ x^2 - 14x + \left(\frac{-14}{2}\right)^2 = -50 + \left(\frac{-14}{2}\right)^2 $$
x squared minus 14 x plus 49 equals negative 50 plus 49 $$ x^2 - 14x + 49 = -50 + 49 $$
Then factor the left side and solve:open parenthesis x minus 7 close parenthesis squared equals negative 1
$$ (x-7)^2 = -1 $$
x minus 7 equals plus or minus the square root of negative 1 $$ x - 7 = \pm\sqrt{-1} $$
x equals 7 plus or minus the square root of negative 1 $$ x = 7 \pm \sqrt{-1} $$
x equals 7 plus or minus I $$ x = 7 \pm i $$
