Algebra Examples

$24 x^{2} - 30 x + 8 = - 2 x$

Step 1

Add $2 x$ to both sides of the equation.

$24 x^{2} - 30 x + 8 + 2 x = 0$

Step 2

Add $- 30 x$ and $2 x$ .

$24 x^{2} - 28 x + 8 = 0$

Step 3

Factor $4$ out of $24 x^{2} - 28 x + 8$ .

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Step 3.1

Factor $4$ out of $24 x^{2}$ .

$4 (6 x^{2}) - 28 x + 8 = 0$

Step 3.2

Factor $4$ out of $- 28 x$ .

$4 (6 x^{2}) + 4 (- 7 x) + 8 = 0$

Step 3.3

Factor $4$ out of $8$ .

$4 (6 x^{2}) + 4 (- 7 x) + 4 (2) = 0$

Step 3.4

Factor $4$ out of $4 (6 x^{2}) + 4 (- 7 x)$ .

$4 (6 x^{2} - 7 x) + 4 (2) = 0$

Step 3.5

Factor $4$ out of $4 (6 x^{2} - 7 x) + 4 (2)$ .

$4 (6 x^{2} - 7 x + 2) = 0$

Step 4

Factor.

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Step 4.1

Factor by grouping.

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Step 4.1.1

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 $a \cdot c = 6 \cdot 2 = 12$ and whose sum is $b = - 7$ .

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Step 4.1.1.1

Factor $- 7$ out of $- 7 x$ .

$4 (6 x^{2} - 7 x + 2) = 0$

Step 4.1.1.2

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

$4 (6 x^{2} + (- 3 - 4) x + 2) = 0$

Step 4.1.1.3

Apply the distributive property.

$4 (6 x^{2} - 3 x - 4 x + 2) = 0$

Step 4.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1

Group the first two terms and the last two terms.

$4 ((6 x^{2} - 3 x) - 4 x + 2) = 0$

Step 4.1.2.2

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

$4 (3 x (2 x - 1) - 2 (2 x - 1)) = 0$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$4 ((2 x - 1) (3 x - 2)) = 0$

Step 4.2

Remove unnecessary parentheses.

$4 (2 x - 1) (3 x - 2) = 0$

Step 5

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

$2 x - 1 = 0$

$3 x - 2 = 0$

Step 6

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 6.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 6.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 7

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

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Step 7.1

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

$3 x - 2 = 0$

Step 7.2

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

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Step 7.2.1

Add $2$ to both sides of the equation.

$3 x = 2$

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 8

The final solution is all the values that make $4 (2 x - 1) (3 x - 2) = 0$ true.

$x = \frac{1}{2}, \frac{2}{3}$