Find the discriminant of the following quadratic expressions.

1. Calculate the discriminant using the formula $b^2 - 4ac$

For $x^2 + x + 1$, the coefficients are $a = 1$, $b = 1$, and $c = 1$.
$\text{Discriminant} = b^2 - 4ac$
$= (1)^2 - 4(1)(1)$
$= 1 - 4$
$= -3$

1. Identify the coefficients of the quadratic expression and substitute them into the discriminant formula.

For $x^2 + 4x - 5$, we have $a = 1$, $b = 4$, and $c = -5$.
$\text{Discriminant} = b^2 - 4ac$
$= (4)^2 - 4(1)(-5)$

2. Evaluate the expression to find the numerical value of the discriminant.

$= 16 + 20$
$= 36$

1. Identify the coefficients $a, b,$ and $c$ of the quadratic expression

For the expression $-x^2 + x - 1$, we have:
$a = -1$
$b = 1$
$c = -1$

2. Calculate the discriminant using the formula $b^2 - 4ac$

$\text{Discriminant} = b^2 - 4ac$
$= (1)^2 - 4(-1)(-1)$
$= 1 - 4$
$= -3$

1. Identify the coefficients of the quadratic expression and substitute into the discriminant formula

The quadratic expression is of the form $ax^2 + bx + c$, where $a = -2$, $b = -4$, and $c = 3$.

Discriminant $= b^2 - 4ac$
$= (-4)^2 - 4(-2)(3)$

2. Evaluate the arithmetic expression to find the final value

$= 16 - (-24)$
$= 16 + 24$
$= 40$

1. Identify the coefficients $a$, $b$, and $c$ from the given quadratic expression

For the quadratic expression $\frac{x^2}{2} + x - \frac{5}{9}$, we can rewrite it as $\frac{1}{2}x^2 + 1x - \frac{5}{9}$.
Therefore, the coefficients are:
$a = \frac{1}{2}$
$b = 1$
$c = -\frac{5}{9}$

2. Substitute the coefficients into the discriminant formula $b^2 - 4ac$

$\text{Discriminant} = b^2 - 4ac$
$= (1)^2 - 4\left(\frac{1}{2}\right)\left(-\frac{5}{9}\right)$

3. Evaluate the expression to find the final value

$= 1 - 2\left(-\frac{5}{9}\right)$
$= 1 + \frac{10}{9}$
$= \frac{19}{9}$

1. Identify the coefficients $a$, $b$, and $c$ of the quadratic expression

Comparing $-x^2 + \frac{x}{3} + 6$ with $ax^2 + bx + c$, we have:
$a = -1$
$b = \frac{1}{3}$
$c = 6$

2. Substitute the coefficients into the discriminant formula and evaluate

$\text{Discriminant} = b^2 - 4ac$
$= \left(\frac{1}{3}\right)^2 - 4(-1)(6)$
$= \frac{1}{9} - (-24)$
$= \frac{1}{9} + 24$
$= \frac{217}{9}$