Algebra Examples

$3 x + 2 x^{2} = 5 x^{2} - 6$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $5 x^{2}$ from both sides of the equation.

$3 x + 2 x^{2} - 5 x^{2} = - 6$

Step 1.2

Add $6$ to both sides of the equation.

$3 x + 2 x^{2} - 5 x^{2} + 6 = 0$

Step 2

Subtract $5 x^{2}$ from $2 x^{2}$ .

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

Step 3

Factor $3$ out of $3 x - 3 x^{2} + 6$ .

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Step 3.1

Factor $3$ out of $3 x$ .

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

Step 3.2

Factor $3$ out of $- 3 x^{2}$ .

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

Step 3.3

Factor $3$ out of $6$ .

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

Step 3.4

Factor $3$ out of $3 (x) + 3 (- x^{2})$ .

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

Step 3.5

Factor $3$ out of $3 (x - x^{2}) + 3 (2)$ .

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

Step 4

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 5

Factor by grouping.

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Step 5.1

Reorder terms.

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

Step 5.2

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 $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = 1$ .

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Step 5.2.1

Multiply by $1$ .

$3 (- u^{2} + 1 u + 2) = 0$

Step 5.2.2

Rewrite $1$ as $- 1$ plus $2$

$3 (- u^{2} + (- 1 + 2) u + 2) = 0$

Step 5.2.3

Apply the distributive property.

$3 (- u^{2} - 1 u + 2 u + 2) = 0$

Step 5.3

Factor out the greatest common factor from each group.

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Step 5.3.1

Group the first two terms and the last two terms.

$3 ((- u^{2} - 1 u) + 2 u + 2) = 0$

Step 5.3.2

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

$3 (u (- u - 1) - 2 (- u - 1)) = 0$

Step 5.4

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

$3 ((- u - 1) (u - 2)) = 0$

Step 6

Factor.

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Step 6.1

Replace all occurrences of $u$ with $x$ .

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

Step 6.2

Remove unnecessary parentheses.

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

Step 7

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

$- x - 1 = 0$

$x - 2 = 0$

Step 8

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

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Step 8.1

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

$- x - 1 = 0$

Step 8.2

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

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Step 8.2.1

Add $1$ to both sides of the equation.

$- x = 1$

Step 8.2.2

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

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Step 8.2.2.1

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

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

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 8.2.2.2.2

Divide $x$ by $1$ .

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

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 9

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

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Step 9.1

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

$x - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$x = 2$

Step 10

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

$x = - 1, 2$