# What is the discriminant of -9x^2+10x=-2x+4 and what does that mean?

Jul 22, 2015

$0$
It means that there is exactly 1 Real solution for this equation

#### Explanation:

The discriminant of a quadratic equation is b^2 – 4ac. To calculate the discriminant of the equation you provided, we move $- 2 x$ and $4$ to the left, resulting in $- 9 {x}^{2} + 12 x - 4$. To calculate the discriminant of this simplified equation, we use our formula above, but substitute $12$ for $b$, $- 9$ as $a$, and $- 4$ as $c$.

We get this equation: ${\left(12\right)}^{2} - 4 \left(- 9\right) \left(- 4\right)$, which evaluates to $0$

The "meaning" is the result of the discriminant being a component of the quadratic formula for the solution(s) to quadratic equation in the form:
$\textcolor{w h i t e}{\text{XXXX}}$$a {x}^{2} + b x + c = 0$
where the solutions can be determined by:
$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Notice that the discriminant is the component within the square root, and as a result:
"discriminant" { (= 0, " one Real root"), (< 0, " no Real Roots"), (> 0, " two Real roots") :}