Finite Math Examples

$u = (7 x + 1) (2 - 3 x)$

Step 1

Rewrite the equation as $(7 x + 1) (2 - 3 x) = u$ .

$(7 x + 1) (2 - 3 x) = u$

Step 2

Simplify $(7 x + 1) (2 - 3 x)$ .

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Step 2.1

Expand $(7 x + 1) (2 - 3 x)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$7 x (2 - 3 x) + 1 (2 - 3 x) = u$

Step 2.1.2

Apply the distributive property.

$7 x \cdot 2 + 7 x (- 3 x) + 1 (2 - 3 x) = u$

Step 2.1.3

Apply the distributive property.

$7 x \cdot 2 + 7 x (- 3 x) + 1 \cdot 2 + 1 (- 3 x) = u$

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $7$ .

$14 x + 7 x (- 3 x) + 1 \cdot 2 + 1 (- 3 x) = u$

Step 2.2.1.2

Rewrite using the commutative property of multiplication.

$14 x + 7 \cdot - 3 x \cdot x + 1 \cdot 2 + 1 (- 3 x) = u$

Step 2.2.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.3.1

Move $x$ .

$14 x + 7 \cdot - 3 (x \cdot x) + 1 \cdot 2 + 1 (- 3 x) = u$

Step 2.2.1.3.2

Multiply $x$ by $x$ .

$14 x + 7 \cdot - 3 x^{2} + 1 \cdot 2 + 1 (- 3 x) = u$

Step 2.2.1.4

Multiply $7$ by $- 3$ .

$14 x - 21 x^{2} + 1 \cdot 2 + 1 (- 3 x) = u$

Step 2.2.1.5

Multiply $2$ by $1$ .

$14 x - 21 x^{2} + 2 + 1 (- 3 x) = u$

Step 2.2.1.6

Multiply $- 3 x$ by $1$ .

$14 x - 21 x^{2} + 2 - 3 x = u$

Step 2.2.2

Subtract $3 x$ from $14 x$ .

$11 x - 21 x^{2} + 2 = u$

Step 3

Subtract $u$ from both sides of the equation.

$11 x - 21 x^{2} + 2 - u = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = - 21$ , $b = 11$ , and $c = 2 - u$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (- 21 \cdot (2 - u))}}{2 \cdot - 21}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot - 21 \cdot (2 - u)}}{2 \cdot - 21}$

Step 6.1.2

Multiply $- 4$ by $- 21$ .

$x = \frac{- 11 \pm \sqrt{121 + 84 \cdot (2 - u)}}{2 \cdot - 21}$

Step 6.1.3

Apply the distributive property.

$x = \frac{- 11 \pm \sqrt{121 + 84 \cdot 2 + 84 (- u)}}{2 \cdot - 21}$

Step 6.1.4

Multiply $84$ by $2$ .

$x = \frac{- 11 \pm \sqrt{121 + 168 + 84 (- u)}}{2 \cdot - 21}$

Step 6.1.5

Multiply $- 1$ by $84$ .

$x = \frac{- 11 \pm \sqrt{121 + 168 - 84 u}}{2 \cdot - 21}$

Step 6.1.6

Add $121$ and $168$ .

$x = \frac{- 11 \pm \sqrt{289 - 84 u}}{2 \cdot - 21}$

Step 6.2

Multiply $2$ by $- 21$ .

$x = \frac{- 11 \pm \sqrt{289 - 84 u}}{- 42}$

Step 6.3

Simplify $\frac{- 11 \pm \sqrt{289 - 84 u}}{- 42}$ .

$x = \frac{11 \pm \sqrt{289 - 84 u}}{42}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{11 + \sqrt{289 - 84 u}}{42}$

$x = \frac{11 - \sqrt{289 - 84 u}}{42}$