Finite Math Examples

$7 (1 - R) = (4 - 4 R^{2})$

Step 1

Simplify $7 (1 - R)$ .

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Step 1.1

Rewrite.

$0 + 0 + 7 (1 - R) = 4 - 4 R^{2}$

Step 1.2

Simplify by adding zeros.

$7 (1 - R) = 4 - 4 R^{2}$

Step 1.3

Apply the distributive property.

$7 \cdot 1 + 7 (- R) = 4 - 4 R^{2}$

Step 1.4

Multiply.

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Step 1.4.1

Multiply $7$ by $1$ .

$7 + 7 (- R) = 4 - 4 R^{2}$

Step 1.4.2

Multiply $- 1$ by $7$ .

$7 - 7 R = 4 - 4 R^{2}$

Step 2

Add $4 R^{2}$ to both sides of the equation.

$7 - 7 R + 4 R^{2} = 4$

Step 3

Subtract $4$ from both sides of the equation.

$7 - 7 R + 4 R^{2} - 4 = 0$

Step 4

Subtract $4$ from $7$ .

$- 7 R + 4 R^{2} + 3 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Let $u = R$ . Substitute $u$ for all occurrences of $R$ .

$- 7 u + 4 u^{2} + 3 = 0$

Step 5.2

Factor by grouping.

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Step 5.2.1

Reorder terms.

$4 u^{2} - 7 u + 3 = 0$

Step 5.2.2

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 $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = - 7$ .

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Step 5.2.2.1

Factor $- 7$ out of $- 7 u$ .

$4 u^{2} - 7 u + 3 = 0$

Step 5.2.2.2

Rewrite $- 7$ as $- 3$ plus $- 4$

$4 u^{2} + (- 3 - 4) u + 3 = 0$

Step 5.2.2.3

Apply the distributive property.

$4 u^{2} - 3 u - 4 u + 3 = 0$

Step 5.2.3

Factor out the greatest common factor from each group.

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Step 5.2.3.1

Group the first two terms and the last two terms.

$(4 u^{2} - 3 u) - 4 u + 3 = 0$

Step 5.2.3.2

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

$u (4 u - 3) - (4 u - 3) = 0$

Step 5.2.4

Factor the polynomial by factoring out the greatest common factor, $4 u - 3$ .

$(4 u - 3) (u - 1) = 0$

Step 5.3

Replace all occurrences of $u$ with $R$ .

$(4 R - 3) (R - 1) = 0$

Step 6

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

$4 R - 3 = 0$

$R - 1 = 0$

Step 7

Set $4 R - 3$ equal to $0$ and solve for $R$ .

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Step 7.1

Set $4 R - 3$ equal to $0$ .

$4 R - 3 = 0$

Step 7.2

Solve $4 R - 3 = 0$ for $R$ .

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Step 7.2.1

Add $3$ to both sides of the equation.

$4 R = 3$

Step 7.2.2

Divide each term in $4 R = 3$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 R = 3$ by $4$ .

$\frac{4 R}{4} = \frac{3}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 R}{4} = \frac{3}{4}$

Step 7.2.2.2.1.2

Divide $R$ by $1$ .

$R = \frac{3}{4}$

Step 8

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

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Step 8.1

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

$R - 1 = 0$

Step 8.2

Add $1$ to both sides of the equation.

$R = 1$

Step 9

The final solution is all the values that make $(4 R - 3) (R - 1) = 0$ true.

$R = \frac{3}{4}, 1$

Step 10

The result can be shown in multiple forms.

Exact Form:

$R = \frac{3}{4}, 1$

Decimal Form:

$R = 0.75, 1$