求下列二次表达式的判别式。

(a)
$x^2 + x + 1$
(b)
$x^2 + 4x - 5$
(c)
$-x^2 + x - 1$
(d)
$-2x^2 - 4x + 3$
(e)
$\frac{x^2}{2} + x - \frac{5}{9}$
(f)
$-x^2 + \frac{x}{3} + 6$

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完整解法

1(a)

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

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

答案

$-3$

1(b)

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.

$= 16 + 20$
$= 36$

答案

$36$

1(c)

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 $b^2 - 4ac$

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

答案

$-3$

1(d)

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

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

答案

1(e)

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

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

答案

$\frac{19}{9}$

1(f)

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

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

答案

$\frac{217}{9}$

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