Question

The function f is defined by f(x) = x2 - 6x +21.
What are the solutions of f(x) = 0?
Show your work on the scratchpad.

Answer 1

The solutions are:

$x =3+2\sqrt{3}i\\\\\:x=3-2\sqrt{3}i$

Solution:

$f(x) = x^2 - 6x + 21$

We have to find solutions when f(x) = 0

$x^2 - 6x + 21 = 0$

Solve by quadratic formula

$\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\\mathrm{For\:}\quad a=1,\:b=-6,\:c=21\\\\x=\frac{-\left(-6\right)\pm \sqrt{\left(-6\right)^2-4\cdot \:1\cdot \:21}}{2\cdot \:1}$

$x = \frac{6 \pm \sqrt{\left(-6\right)^2-4\cdot \:1\cdot \:21}}{2\cdot \:1}\\\\Simplify\\\\x = \frac{6 \pm \sqrt{36 - 84}}{2}\\\\x = \frac{6 \pm \sqrt{-48}}{2}\\\\x = \frac{6 \pm i \sqrt{48}}{2}\\\\simplify\\\\x = \frac{6 \pm 4\sqrt{3}i}{2}$

$x = 3 \pm 2 \sqrt{3} i$

We have two solutions:

$x = 3 + 2 \sqrt{3} i\\\\x = 3 - 2 \sqrt{3} i$

Thus the solutions are:

$x =3+2\sqrt{3}i\\\\\:x=3-2\sqrt{3}i$