Algebra Examples

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

Step 1

Move all terms containing $x$ 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

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

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

Step 2

Add $6$ to both sides of the equation.

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

Step 3

Factor the left side of the equation.

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Step 3.1

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

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Step 3.1.1

Reorder $3 x$ and $- 3 x^{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Factor $- 3$ out of $3 x$ .

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

Step 3.1.4

Factor $- 3$ out of $6$ .

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} - x - 2$ using the AC method.

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Step 3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 3.2.1.2

Write the factored form using these integers.

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

Step 3.2.2

Remove unnecessary parentheses.

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

Step 4

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

$x - 2 = 0$

$x + 1 = 0$

Step 5

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

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Step 5.1

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

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

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Step 6.1

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

$x + 1 = 0$

Step 6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

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

$x = 2, - 1$