Answer:
[tex]b = -2[/tex]
[tex]c=\dfrac{51}{2}[/tex]
Step-by-step explanation:
The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0 by providing the values of the variable x.
[tex]\boxed{\begin{array}{l}\underline{\sf Quadratic\;Formula}\\\\x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\\\\\textsf{when} \;ax^2+bx+c=0 \\\end{array}}[/tex]
The given roots of a quadratic equation are:
[tex]x=\dfrac{1+i\sqrt{50}}{2}\;\;\textsf{and}\;\;x=\dfrac{1-i\sqrt{50}}{2}[/tex]
To find the values of b and c if a = 2, begin by substituting a = 2 into the quadratic formula:
[tex]x=\dfrac{-b \pm \sqrt{b^2-4(2)c}}{2(2)}\\\\\\\\x=\dfrac{-b \pm \sqrt{b^2-8c}}{4}\\\\\\[/tex]
Now equate this to the given roots:
[tex]\dfrac{-b \pm \sqrt{b^2-8c}}{4}=\dfrac{1\pm i\sqrt{50}}{2}[/tex]
Multiply the numerator and denominator of the right side of the equation by 2:
[tex]\dfrac{-b \pm \sqrt{b^2-8c}}{4}=\dfrac{2(1\pm i\sqrt{50})}{2(2)}\\\\\\\\\dfrac{-b \pm \sqrt{b^2-8c}}{4}=\dfrac{2\pm i\:2\sqrt{50}}{4}[/tex]
Multiply both sides by 4 to cancel the denominators:
[tex]-b\pm \sqrt{b^2-8c}=2\pm i\:2\sqrt{50}[/tex]
Comparing both sides of the equation, the value of b is:
[tex]-b = 2 \\\\ b = -2[/tex]
Substitute b = -2 into the equation:
[tex]-(-2)\pm \sqrt{(-2)^2-8c}=2\pm i\:2\sqrt{50}\\\\\\2\pm\sqrt{4-8c}=2\pm i\:2\sqrt{50}[/tex]
Subtract 2 from both sides:
[tex]\pm\sqrt{4-8c}=\pm i\:2\sqrt{50}\\\\\\\sqrt{4-8c}= i\:2\sqrt{50}[/tex]
Square both sides:
[tex]4-8c=i^2\left(2\sqrt{50}\right)^2\\\\\\4-8c=200\:i^2[/tex]
The square of the imaginary unit [tex]i[/tex] is equal to -1:
[tex]4-8c=200 \cdot (-1)\\\\\\4-8c=-200[/tex]
Solve for c:
[tex]-8c=-204\\\\\\c=\dfrac{-204}{-8}\\\\\\c=\dfrac{51}{2}[/tex]
Therefore, the value of c is 51/2.
