Trigonometry Examples

$12 x^{2} + 7 x - 10 = 0$

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot - 10 = - 120$ and whose sum is $b = 7$ .

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Step 1.1.1

Factor $7$ out of $7 x$ .

$12 x^{2} + 7 (x) - 10 = 0$

Step 1.1.2

Rewrite $7$ as $- 8$ plus $15$

$12 x^{2} + (- 8 + 15) x - 10 = 0$

Step 1.1.3

Apply the distributive property.

$12 x^{2} - 8 x + 15 x - 10 = 0$

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.

$(12 x^{2} - 8 x) + 15 x - 10 = 0$

Step 1.2.2

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

$4 x (3 x - 2) + 5 (3 x - 2) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(3 x - 2) (4 x + 5) = 0$

Step 2

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

$3 x - 2 = 0$

$4 x + 5 = 0$

Step 3

Set $3 x - 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Step 3.2

Solve $3 x - 2 = 0$ for $x$ .

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Step 3.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 3.2.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 4

Set $4 x + 5$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $4 x + 5$ equal to $0$ .

$4 x + 5 = 0$

Step 4.2

Solve $4 x + 5 = 0$ for $x$ .

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Step 4.2.1

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Step 4.2.2

Divide each term in $4 x = - 5$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

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.

$x = - \frac{5}{4}$

Step 5

The final solution is all the values that make $(3 x - 2) (4 x + 5) = 0$ true.

$x = \frac{2}{3}, - \frac{5}{4}$