Trigonometry Examples

$3 t^{2} = 7 t - 4$

Step 1

Subtract $7 t$ from both sides of the equation.

$3 t^{2} - 7 t = - 4$

Step 2

Add $4$ to both sides of the equation.

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

Step 3

Factor by grouping.

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Step 3.1

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

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Step 3.1.1

Factor $- 7$ out of $- 7 t$ .

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $t - 1$ .

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

Step 4

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

$t - 1 = 0$

$3 t - 4 = 0$

Step 5

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

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Step 5.1

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

$t - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$t = 1$

Step 6

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

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Step 6.1

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

$3 t - 4 = 0$

Step 6.2

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

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Step 6.2.1

Add $4$ to both sides of the equation.

$3 t = 4$

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $t$ by $1$ .

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

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$t = 1, 1.\overline{3}$

Mixed Number Form:

$t = 1, 1 \frac{1}{3}$