Pre-Algebra Examples

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

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 1.1.1

Multiply by $1$ .

$3 x^{2} + 1 x - 4 = 0$

Step 1.1.2

Rewrite $1$ as $- 3$ plus $4$

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

Step 1.1.3

Apply the distributive property.

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

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.

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

Step 1.2.2

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

$3 x (x - 1) + 4 (x - 1) = 0$

Step 1.3

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

$(x - 1) (3 x + 4) = 0$

Step 2

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

$x - 1 = 0$

$3 x + 4 = 0$

Step 3

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

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Step 3.1

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

$x - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Set $3 x + 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Step 4.2

Solve $3 x + 4 = 0$ for $x$ .

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 4.2.2

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

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Step 4.2.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

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.

$x = - \frac{4}{3}$

Step 5

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

$x = 1, - \frac{4}{3}$