Finite Math Examples

$4 r^{2} - 6 r + 4 = 2$

Step 1

Subtract $2$ from both sides of the equation.

$4 r^{2} - 6 r + 4 - 2 = 0$

Step 2

Subtract $2$ from $4$ .

$4 r^{2} - 6 r + 2 = 0$

Step 3

Factor $2$ out of $4 r^{2} - 6 r + 2$ .

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Step 3.1

Factor $2$ out of $4 r^{2}$ .

$2 (2 r^{2}) - 6 r + 2 = 0$

Step 3.2

Factor $2$ out of $- 6 r$ .

$2 (2 r^{2}) + 2 (- 3 r) + 2 = 0$

Step 3.3

Factor $2$ out of $2$ .

$2 (2 r^{2}) + 2 (- 3 r) + 2 (1) = 0$

Step 3.4

Factor $2$ out of $2 (2 r^{2}) + 2 (- 3 r)$ .

$2 (2 r^{2} - 3 r) + 2 (1) = 0$

Step 3.5

Factor $2$ out of $2 (2 r^{2} - 3 r) + 2 (1)$ .

$2 (2 r^{2} - 3 r + 1) = 0$

Step 4

Factor.

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Step 4.1

Factor by grouping.

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Step 4.1.1

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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Step 4.1.1.1

Factor $- 3$ out of $- 3 r$ .

$2 (2 r^{2} - 3 r + 1) = 0$

Step 4.1.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

$2 (2 r^{2} + (- 1 - 2) r + 1) = 0$

Step 4.1.1.3

Apply the distributive property.

$2 (2 r^{2} - 1 r - 2 r + 1) = 0$

Step 4.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1

Group the first two terms and the last two terms.

$2 ((2 r^{2} - 1 r) - 2 r + 1) = 0$

Step 4.1.2.2

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

$2 (r (2 r - 1) - (2 r - 1)) = 0$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $2 r - 1$ .

$2 ((2 r - 1) (r - 1)) = 0$

Step 4.2

Remove unnecessary parentheses.

$2 (2 r - 1) (r - 1) = 0$

Step 5

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

$2 r - 1 = 0$

$r - 1 = 0$

Step 6

Set $2 r - 1$ equal to $0$ and solve for $r$ .

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Step 6.1

Set $2 r - 1$ equal to $0$ .

$2 r - 1 = 0$

Step 6.2

Solve $2 r - 1 = 0$ for $r$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$2 r = 1$

Step 6.2.2

Divide each term in $2 r = 1$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 r = 1$ by $2$ .

$\frac{2 r}{2} = \frac{1}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 r}{2} = \frac{1}{2}$

Step 6.2.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{1}{2}$

Step 7

Set $r - 1$ equal to $0$ and solve for $r$ .

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Step 7.1

Set $r - 1$ equal to $0$ .

$r - 1 = 0$

Step 7.2

Add $1$ to both sides of the equation.

$r = 1$

Step 8

The final solution is all the values that make $2 (2 r - 1) (r - 1) = 0$ true.

$r = \frac{1}{2}, 1$