Finite Math Examples

$78 = 2 x^{2} - 20 x + 120$

Step 1

Rewrite the equation as $2 x^{2} - 20 x + 120 = 78$ .

$2 x^{2} - 20 x + 120 = 78$

Step 2

Subtract $78$ from both sides of the equation.

$2 x^{2} - 20 x + 120 - 78 = 0$

Step 3

Subtract $78$ from $120$ .

$2 x^{2} - 20 x + 42 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $2$ out of $2 x^{2} - 20 x + 42$ .

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Step 4.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 20 x + 42 = 0$

Step 4.1.2

Factor $2$ out of $- 20 x$ .

$2 (x^{2}) + 2 (- 10 x) + 42 = 0$

Step 4.1.3

Factor $2$ out of $42$ .

$2 x^{2} + 2 (- 10 x) + 2 \cdot 21 = 0$

Step 4.1.4

Factor $2$ out of $2 x^{2} + 2 (- 10 x)$ .

$2 (x^{2} - 10 x) + 2 \cdot 21 = 0$

Step 4.1.5

Factor $2$ out of $2 (x^{2} - 10 x) + 2 \cdot 21$ .

$2 (x^{2} - 10 x + 21) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} - 10 x + 21$ using the AC method.

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Step 4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 4.2.1.2

Write the factored form using these integers.

$2 ((x - 7) (x - 3)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$2 (x - 7) (x - 3) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 7 = 0$

$x - 3 = 0$

Step 6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 8

The final solution is all the values that make $2 (x - 7) (x - 3) = 0$ true.

$x = 7, 3$