Finite Math Examples

${(6 - 7 x)}^{2} = 6 - 7 x \cdot x$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$6 - 7 x \cdot x = {(6 - 7 x)}^{2}$

Step 2

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

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Step 2.1

Move $x$ .

$6 - 7 (x \cdot x) = {(6 - 7 x)}^{2}$

Step 2.2

Multiply $x$ by $x$ .

$6 - 7 x^{2} = {(6 - 7 x)}^{2}$

Step 3

Simplify ${(6 - 7 x)}^{2}$ .

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Step 3.1

Rewrite ${(6 - 7 x)}^{2}$ as $(6 - 7 x) (6 - 7 x)$ .

$6 - 7 x^{2} = (6 - 7 x) (6 - 7 x)$

Step 3.2

Expand $(6 - 7 x) (6 - 7 x)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$6 - 7 x^{2} = 6 (6 - 7 x) - 7 x (6 - 7 x)$

Step 3.2.2

Apply the distributive property.

$6 - 7 x^{2} = 6 \cdot 6 + 6 (- 7 x) - 7 x (6 - 7 x)$

Step 3.2.3

Apply the distributive property.

$6 - 7 x^{2} = 6 \cdot 6 + 6 (- 7 x) - 7 x \cdot 6 - 7 x (- 7 x)$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $6$ by $6$ .

$6 - 7 x^{2} = 36 + 6 (- 7 x) - 7 x \cdot 6 - 7 x (- 7 x)$

Step 3.3.1.2

Multiply $- 7$ by $6$ .

$6 - 7 x^{2} = 36 - 42 x - 7 x \cdot 6 - 7 x (- 7 x)$

Step 3.3.1.3

Multiply $6$ by $- 7$ .

$6 - 7 x^{2} = 36 - 42 x - 42 x - 7 x (- 7 x)$

Step 3.3.1.4

Rewrite using the commutative property of multiplication.

$6 - 7 x^{2} = 36 - 42 x - 42 x - 7 \cdot - 7 x \cdot x$

Step 3.3.1.5

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

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Step 3.3.1.5.1

Move $x$ .

$6 - 7 x^{2} = 36 - 42 x - 42 x - 7 \cdot - 7 (x \cdot x)$

Step 3.3.1.5.2

Multiply $x$ by $x$ .

$6 - 7 x^{2} = 36 - 42 x - 42 x - 7 \cdot - 7 x^{2}$

Step 3.3.1.6

Multiply $- 7$ by $- 7$ .

$6 - 7 x^{2} = 36 - 42 x - 42 x + 49 x^{2}$

Step 3.3.2

Subtract $42 x$ from $- 42 x$ .

$6 - 7 x^{2} = 36 - 84 x + 49 x^{2}$

Step 4

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$36 - 84 x + 49 x^{2} = 6 - 7 x^{2}$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Add $7 x^{2}$ to both sides of the equation.

$36 - 84 x + 49 x^{2} + 7 x^{2} = 6$

Step 5.2

Add $49 x^{2}$ and $7 x^{2}$ .

$36 - 84 x + 56 x^{2} = 6$

Step 6

Subtract $6$ from both sides of the equation.

$36 - 84 x + 56 x^{2} - 6 = 0$

Step 7

Subtract $6$ from $36$ .

$- 84 x + 56 x^{2} + 30 = 0$

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $2$ out of $- 84 x + 56 x^{2} + 30$ .

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Step 8.1.1

Factor $2$ out of $- 84 x$ .

$2 (- 42 x) + 56 x^{2} + 30 = 0$

Step 8.1.2

Factor $2$ out of $56 x^{2}$ .

$2 (- 42 x) + 2 (28 x^{2}) + 30 = 0$

Step 8.1.3

Factor $2$ out of $30$ .

$2 (- 42 x) + 2 (28 x^{2}) + 2 (15) = 0$

Step 8.1.4

Factor $2$ out of $2 (- 42 x) + 2 (28 x^{2})$ .

$2 (- 42 x + 28 x^{2}) + 2 (15) = 0$

Step 8.1.5

Factor $2$ out of $2 (- 42 x + 28 x^{2}) + 2 (15)$ .

$2 (- 42 x + 28 x^{2} + 15) = 0$

Step 8.2

Reorder terms.

$2 (28 x^{2} - 42 x + 15) = 0$

Step 9

Divide each term in $2 (28 x^{2} - 42 x + 15) = 0$ by $2$ and simplify.

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Step 9.1

Divide each term in $2 (28 x^{2} - 42 x + 15) = 0$ by $2$ .

$\frac{2 (28 x^{2} - 42 x + 15)}{2} = \frac{0}{2}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{2 (28 x^{2} - 42 x + 15)}{2} = \frac{0}{2}$

Step 9.2.1.2

Divide $28 x^{2} - 42 x + 15$ by $1$ .

$28 x^{2} - 42 x + 15 = \frac{0}{2}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $0$ by $2$ .

$28 x^{2} - 42 x + 15 = 0$

Step 10

Use the quadratic formula to find the solutions.

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

Step 11

Substitute the values $a = 28$ , $b = - 42$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{42 \pm \sqrt{{(- 42)}^{2} - 4 \cdot (28 \cdot 15)}}{2 \cdot 28}$

Step 12

Simplify.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Raise $- 42$ to the power of $2$ .

$x = \frac{42 \pm \sqrt{1764 - 4 \cdot 28 \cdot 15}}{2 \cdot 28}$

Step 12.1.2

Multiply $- 4 \cdot 28 \cdot 15$ .

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Step 12.1.2.1

Multiply $- 4$ by $28$ .

$x = \frac{42 \pm \sqrt{1764 - 112 \cdot 15}}{2 \cdot 28}$

Step 12.1.2.2

Multiply $- 112$ by $15$ .

$x = \frac{42 \pm \sqrt{1764 - 1680}}{2 \cdot 28}$

Step 12.1.3

Subtract $1680$ from $1764$ .

$x = \frac{42 \pm \sqrt{84}}{2 \cdot 28}$

Step 12.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 12.1.4.1

Factor $4$ out of $84$ .

$x = \frac{42 \pm \sqrt{4 (21)}}{2 \cdot 28}$

Step 12.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{42 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 28}$

Step 12.1.5

Pull terms out from under the radical.

$x = \frac{42 \pm 2 \sqrt{21}}{2 \cdot 28}$

Step 12.2

Multiply $2$ by $28$ .

$x = \frac{42 \pm 2 \sqrt{21}}{56}$

Step 12.3

Simplify $\frac{42 \pm 2 \sqrt{21}}{56}$ .

$x = \frac{21 \pm \sqrt{21}}{28}$

Step 13

The final answer is the combination of both solutions.

$x = \frac{21 + \sqrt{21}}{28}, \frac{21 - \sqrt{21}}{28}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = \frac{21 + \sqrt{21}}{28}, \frac{21 - \sqrt{21}}{28}$

Decimal Form:

$x = 0.91366341 \dots, 0.58633658 \dots$