Algebra Examples

$3 x^{2} + x - 1 = (2 x + 9) (x - 1)$

Step 1

Simplify $(2 x + 9) (x - 1)$ .

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Step 1.1

Rewrite.

$3 x^{2} + x - 1 = 0 + 0 + (2 x + 9) (x - 1)$

Step 1.2

Simplify by adding zeros.

$3 x^{2} + x - 1 = (2 x + 9) (x - 1)$

Step 1.3

Expand $(2 x + 9) (x - 1)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 x^{2} + x - 1 = 2 x (x - 1) + 9 (x - 1)$

Step 1.3.2

Apply the distributive property.

$3 x^{2} + x - 1 = 2 x \cdot x + 2 x \cdot - 1 + 9 (x - 1)$

Step 1.3.3

Apply the distributive property.

$3 x^{2} + x - 1 = 2 x \cdot x + 2 x \cdot - 1 + 9 x + 9 \cdot - 1$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.1.1

Move $x$ .

$3 x^{2} + x - 1 = 2 (x \cdot x) + 2 x \cdot - 1 + 9 x + 9 \cdot - 1$

Step 1.4.1.1.2

Multiply $x$ by $x$ .

$3 x^{2} + x - 1 = 2 x^{2} + 2 x \cdot - 1 + 9 x + 9 \cdot - 1$

Step 1.4.1.2

Multiply $- 1$ by $2$ .

$3 x^{2} + x - 1 = 2 x^{2} - 2 x + 9 x + 9 \cdot - 1$

Step 1.4.1.3

Multiply $9$ by $- 1$ .

$3 x^{2} + x - 1 = 2 x^{2} - 2 x + 9 x - 9$

Step 1.4.2

Add $- 2 x$ and $9 x$ .

$3 x^{2} + x - 1 = 2 x^{2} + 7 x - 9$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

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

$3 x^{2} + x - 1 - 2 x^{2} = 7 x - 9$

Step 2.2

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

$3 x^{2} + x - 1 - 2 x^{2} - 7 x = - 9$

Step 2.3

Subtract $2 x^{2}$ from $3 x^{2}$ .

$x^{2} + x - 1 - 7 x = - 9$

Step 2.4

Subtract $7 x$ from $x$ .

$x^{2} - 6 x - 1 = - 9$

Step 3

Add $9$ to both sides of the equation.

$x^{2} - 6 x - 1 + 9 = 0$

Step 4

Add $- 1$ and $9$ .

$x^{2} - 6 x + 8 = 0$

Step 5

Factor $x^{2} - 6 x + 8$ using the AC method.

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Step 5.1

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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 5.2

Write the factored form using these integers.

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

Step 6

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

$x - 4 = 0$

$x - 2 = 0$

Step 7

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 7.2

Add $4$ to both sides of the equation.

$x = 4$

Step 8

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

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Step 8.1

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

$x - 2 = 0$

Step 8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 9

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

$x = 4, 2$