Finite Math Examples

$(1 - a) (3 - a) - 8$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(1 - a) (3 - a) - 8$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Expand $(1 - a) (3 - a)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$1 (3 - a) - a (3 - a) - 8 = 0$

Step 1.1.1.1.2

Apply the distributive property.

$1 \cdot 3 + 1 (- a) - a (3 - a) - 8 = 0$

Step 1.1.1.1.3

Apply the distributive property.

$1 \cdot 3 + 1 (- a) - a \cdot 3 - a (- a) - 8 = 0$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Multiply $3$ by $1$ .

$3 + 1 (- a) - a \cdot 3 - a (- a) - 8 = 0$

Step 1.1.1.2.1.2

Multiply $- a$ by $1$ .

$3 - a - a \cdot 3 - a (- a) - 8 = 0$

Step 1.1.1.2.1.3

Multiply $3$ by $- 1$ .

$3 - a - 3 a - a (- a) - 8 = 0$

Step 1.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$3 - a - 3 a - 1 \cdot - 1 a \cdot a - 8 = 0$

Step 1.1.1.2.1.5

Multiply $a$ by $a$ by adding the exponents.

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Step 1.1.1.2.1.5.1

Move $a$ .

$3 - a - 3 a - 1 \cdot - 1 (a \cdot a) - 8 = 0$

Step 1.1.1.2.1.5.2

Multiply $a$ by $a$ .

$3 - a - 3 a - 1 \cdot - 1 a^{2} - 8 = 0$

Step 1.1.1.2.1.6

Multiply $- 1$ by $- 1$ .

$3 - a - 3 a + 1 a^{2} - 8 = 0$

Step 1.1.1.2.1.7

Multiply $a^{2}$ by $1$ .

$3 - a - 3 a + a^{2} - 8 = 0$

Step 1.1.1.2.2

Subtract $3 a$ from $- a$ .

$3 - 4 a + a^{2} - 8 = 0$

Step 1.1.2

Subtract $8$ from $3$ .

$- 4 a + a^{2} - 5 = 0$

Step 2

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 3

Substitute in the values of $a$ , $b$ , and $c$ .

${(- 4)}^{2} - 4 (1 \cdot - 5)$

Step 4

Evaluate the result to find the discriminant.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 4$ to the power of $2$ .

$16 - 4 (1 \cdot - 5)$

Step 4.1.2

Multiply $- 4 (1 \cdot - 5)$ .

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Step 4.1.2.1

Multiply $- 5$ by $1$ .

$16 - 4 \cdot - 5$

Step 4.1.2.2

Multiply $- 4$ by $- 5$ .

$16 + 20$

Step 4.2

Add $16$ and $20$ .

$36$