# How do you find the discriminant and how many and what type of solutions does x^2 + 5x + 7 = 0 have?

##### 2 Answers
Mar 10, 2018

See a solution process below:

#### Explanation:

The quadratic formula states:

For $a {x}^{2} + b x + c = 0$, the values of $x$ which are the solutions to the equation are given by:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The discriminate is the portion of the quadratic equation within the radical: ${\textcolor{b l u e}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}$

If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions

To find the discriminant for this problem substitute:

$\textcolor{red}{1}$ for $\textcolor{red}{a}$

$\textcolor{b l u e}{5}$ for $\textcolor{b l u e}{b}$

$\textcolor{g r e e n}{7}$ for $\textcolor{g r e e n}{c}$

Giving

${\textcolor{b l u e}{5}}^{2} - \left(4 \cdot \textcolor{red}{1} \cdot \textcolor{g r e e n}{7}\right) \implies$

$25 - 28 \implies$

$- 3$

Because the number is negative you will get two complex solutions.

Mar 10, 2018

${x}^{2} + 5 x + 7 = 0$ has no real roots therefore no defined solutions.

#### Explanation:

D = ${b}^{2} - 4 a c$

A standard polynomial form is:
$a {x}^{2} + b x + c = 0$

From your equation:
a= 1
b=5
c=7

∴ 25 - 4 x 1 x 7
= -3

A negative discriminant means there is no real root.
If you wanted to go further, it would have complex x roots. But yeah:

${x}^{2} + 5 x + 7 = 0$ has no real roots therefore no defined solutions.