Trigonometry Examples

7 j^{2} + 2 j - 9 = 0

$7 j^{2} + 2 j - 9 = 0$

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 7 \cdot - 9 = - 63

$a \cdot c = 7 \cdot - 9 = - 63$ and whose sum is

b = 2

$b = 2$ .

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Step 1.1.1

Factor

2

$2$ out of

2 j

$2 j$ .

7 j^{2} + 2 (j) - 9 = 0

$7 j^{2} + 2 (j) - 9 = 0$

Step 1.1.2

Rewrite

2

$2$ as

- 7

$- 7$ plus

9

$9$

7 j^{2} + (- 7 + 9) j - 9 = 0

$7 j^{2} + (- 7 + 9) j - 9 = 0$

Step 1.1.3

Apply the distributive property.

7 j^{2} - 7 j + 9 j - 9 = 0

$7 j^{2} - 7 j + 9 j - 9 = 0$

7 j^{2} - 7 j + 9 j - 9 = 0

$7 j^{2} - 7 j + 9 j - 9 = 0$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

(7 j^{2} - 7 j) + 9 j - 9 = 0

$(7 j^{2} - 7 j) + 9 j - 9 = 0$

Step 1.2.2

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

7 j (j - 1) + 9 (j - 1) = 0

$7 j (j - 1) + 9 (j - 1) = 0$

7 j (j - 1) + 9 (j - 1) = 0

$7 j (j - 1) + 9 (j - 1) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor,

j - 1

$j - 1$ .

(j - 1) (7 j + 9) = 0

$(j - 1) (7 j + 9) = 0$

(j - 1) (7 j + 9) = 0

$(j - 1) (7 j + 9) = 0$

Step 2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

j - 1 = 0

$j - 1 = 0$

7 j + 9 = 0

$7 j + 9 = 0$

Step 3

Set

j - 1

$j - 1$ equal to

0

$0$ and solve for

j

$j$ .

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Step 3.1

Set

j - 1

$j - 1$ equal to

0

$0$ .

j - 1 = 0

$j - 1 = 0$

Step 3.2

Add

1

$1$ to both sides of the equation.

j = 1

$j = 1$

j = 1

$j = 1$

Step 4

Set

7 j + 9

$7 j + 9$ equal to

0

$0$ and solve for

j

$j$ .

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Step 4.1

Set

7 j + 9

$7 j + 9$ equal to

0

$0$ .

7 j + 9 = 0

$7 j + 9 = 0$

Step 4.2

Solve

7 j + 9 = 0

$7 j + 9 = 0$ for

j

$j$ .

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Step 4.2.1

Subtract

9

$9$ from both sides of the equation.

7 j = - 9

$7 j = - 9$

Step 4.2.2

Divide each term in

7 j = - 9

$7 j = - 9$ by

7

$7$ and simplify.

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Step 4.2.2.1

Divide each term in

7 j = - 9

$7 j = - 9$ by

7

$7$ .

\frac{7 j}{7} = \frac{- 9}{7}

$\frac{7 j}{7} = \frac{- 9}{7}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of

7

$7$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{7 j}{7} = \frac{- 9}{7}$

Step 4.2.2.2.1.2

Divide

$j$ by

$1$ .

$j = \frac{- 9}{7}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$j = - \frac{9}{7}$

Step 5

The final solution is all the values that make

$(j - 1) (7 j + 9) = 0$ true.

$j = 1, - \frac{9}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$j = 1, - \frac{9}{7}$

Decimal Form:

$j = 1, - 1.\overline{285714}$

Mixed Number Form:

$j = 1, - 1 \frac{2}{7}$