# How do you find the discriminant and how many solutions does -7d^2 + 2d + 9 = 0 have?

May 11, 2015

The equation is of the form color(blue)(ax^2+bx+c=0 where:

$a = - 7 , b = 2 , c = 9$

The Disciminant is given by :
$\Delta = {b}^{2} - 4 \cdot a \cdot c$
$= {\left(2\right)}^{2} - \left(4 \cdot \left(- 7\right) \cdot 9\right)$
$= 4 - \left(- 252\right) = 4 + 252 = 256$

If $\Delta = 0$ then there is only one solution.
(for $\Delta > 0$ there are two solutions,
for $\Delta < 0$ there are no real solutions)

As $\Delta = 256$, this equation has TWO REAL SOLUTIONS

• Note :
The solutions are normally found using the formula
$x = \frac{- b \pm \sqrt{\Delta}}{2 \cdot a}$

As $\Delta = 256$, $x = \frac{- \left(2\right) \pm \sqrt{256}}{2 \cdot - 7} = \frac{- 2 \pm 16}{- 14} = \frac{14}{-} 14 \mathmr{and} \frac{- 18}{-} 14 = - 1 \mathmr{and} \frac{9}{7}$

color(green)(x = -1,9/7 are the two solutions