Finite Math Examples

$7 x^{2} = 22 x - 3$

Step 1

Subtract $22 x$ from both sides of the equation.

$7 x^{2} - 22 x = - 3$

Step 2

Add $3$ to both sides of the equation.

$7 x^{2} - 22 x + 3 = 0$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot 3 = 21$ and whose sum is $b = - 22$ .

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Step 3.1.1

Factor $- 22$ out of $- 22 x$ .

$7 x^{2} - 22 x + 3 = 0$

Step 3.1.2

Rewrite $- 22$ as $- 1$ plus $- 21$

$7 x^{2} + (- 1 - 21) x + 3 = 0$

Step 3.1.3

Apply the distributive property.

$7 x^{2} - 1 x - 21 x + 3 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(7 x^{2} - 1 x) - 21 x + 3 = 0$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (7 x - 1) - 3 (7 x - 1) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $7 x - 1$ .

$(7 x - 1) (x - 3) = 0$

Step 4

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

$7 x - 1 = 0$

$x - 3 = 0$

Step 5

Set $7 x - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $7 x - 1$ equal to $0$ .

$7 x - 1 = 0$

Step 5.2

Solve $7 x - 1 = 0$ for $x$ .

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Step 5.2.1

Add $1$ to both sides of the equation.

$7 x = 1$

Step 5.2.2

Divide each term in $7 x = 1$ by $7$ and simplify.

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Step 5.2.2.1

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Step 6

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

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Step 6.1

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

$x - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7

The final solution is all the values that make $(7 x - 1) (x - 3) = 0$ true.

$x = \frac{1}{7}, 3$

Step 8