# What is the discriminant of x^2 - 5x = 6 and what does that mean?

Aug 2, 2015

$\Delta = 49$

#### Explanation:

For a quadratic equation that has the general form

$\textcolor{b l u e}{a {x}^{2} + b x + c = 0}$

the discriminant can be calculated by the formula

$\textcolor{b l u e}{\Delta = {b}^{2} - 4 \cdot a \cdot c}$

Rearrange your quadratic by adding $- 6$ to both sides of the equation

${x}^{2} - 5 x - 6 = \textcolor{red}{\cancel{\textcolor{b l a c k}{6}}} - \textcolor{red}{\cancel{\textcolor{b l a c k}{6}}}$

${x}^{2} - 5 x - 6 = 0$

In your case, you have $a = 1$, $b = - 5$, and $c = - 6$, so the discriminant will be equal to

$\Delta = {\left(- 5\right)}^{2} - 4 \cdot 1 \cdot \left(- 6\right)$

$\Delta = 25 + 24 = 49$

SInce $\Delta > 0$, this quadratic equation will have two disctinct real solutions. Moreover, because $\Delta$ is a perfect square, those two solutions will be rational numbers.

The general form of the two solutions is given by the quadratic formula

color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a)

In your case, these two solutions will be

${x}_{1 , 2} = \frac{- \left(- 5\right) \pm \sqrt{49}}{2 \cdot 1} = \frac{5 \pm 7}{2}$

so that

${x}_{1} = \frac{5 + 7}{2} = \textcolor{g r e e n}{6}$ and ${x}_{2} = \frac{5 - 7}{2} = \textcolor{g r e e n}{- 1}$

Aug 2, 2015

Solve: ${x}^{2} - 5 x = 6$

#### Explanation:

$y = {x}^{2} - 5 x - 6 = 0$
In this case, (a - b + c = 0), use the shortcut --> 2 real roots--> - 1 and $\left(- \frac{c}{a} = 6\right) .$

REMINDER of SHORCUT

When (a + b + c = 0) --> 2 real roots: 1 and $\frac{c}{a}$
When (a - b + c = 0) --> 2 real roots: - 1 and $\left(- \frac{c}{a}\right)$
Remember this shortcut. It will save you a lot of time and effort.