How do you solve using the quadratic formula x^2-4x+10=0?

May 3, 2015

Given a quadratic equation of the form
$a {x}^{2} + b x + c = 0$
the quadratic roots can be evaluated using the formula
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

For ${x}^{2} - 4 x + 10 = 0$

$x = \frac{4 \pm \sqrt{16 - 40}}{2}$

$x = 2 \pm \sqrt{- 6}$

This equation has no Real solutions
but if we are allowed Complex solutions:
$x = 2 \pm \sqrt{6} i$

May 4, 2015

You notice the 1st degree coefficient is even, so you can use the direct formula
P(x)=ax^2+bx+c, β=b2
x1,x2={−β±sqrt(β^2−ac)}/a
so x_1,x_2=2±sqrt(4−10)=2±sqrt(−6)
So you know there are no solutions in ℝ, and you have the complex conjugated solutions $z = 2 + \sqrt{6} i$ andbarz=2−sqrt(6)i