Algebra Examples

$3 {(x + 4)}^{2} = 10 x + 32$

Step 1

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

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Step 1.1

Subtract $10 x$ from both sides of the equation.

$3 {(x + 4)}^{2} - 10 x = 32$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$3 ((x + 4) (x + 4)) - 10 x = 32$

Step 1.2.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$3 (x (x + 4) + 4 (x + 4)) - 10 x = 32$

Step 1.2.2.2

Apply the distributive property.

$3 (x \cdot x + x \cdot 4 + 4 (x + 4)) - 10 x = 32$

Step 1.2.2.3

Apply the distributive property.

$3 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) - 10 x = 32$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $x$ by $x$ .

$3 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) - 10 x = 32$

Step 1.2.3.1.2

Move $4$ to the left of $x$ .

$3 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) - 10 x = 32$

Step 1.2.3.1.3

Multiply $4$ by $4$ .

$3 (x^{2} + 4 x + 4 x + 16) - 10 x = 32$

Step 1.2.3.2

Add $4 x$ and $4 x$ .

$3 (x^{2} + 8 x + 16) - 10 x = 32$

Step 1.2.4

Apply the distributive property.

$3 x^{2} + 3 (8 x) + 3 \cdot 16 - 10 x = 32$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Multiply $8$ by $3$ .

$3 x^{2} + 24 x + 3 \cdot 16 - 10 x = 32$

Step 1.2.5.2

Multiply $3$ by $16$ .

$3 x^{2} + 24 x + 48 - 10 x = 32$

Step 1.3

Subtract $10 x$ from $24 x$ .

$3 x^{2} + 14 x + 48 = 32$

Step 2

Subtract $32$ from both sides of the equation.

$3 x^{2} + 14 x + 48 - 32 = 0$

Step 3

Subtract $32$ from $48$ .

$3 x^{2} + 14 x + 16 = 0$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 3 \cdot 16 = 48$ and whose sum is $b = 14$ .

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Step 4.1.1

Factor $14$ out of $14 x$ .

$3 x^{2} + 14 (x) + 16 = 0$

Step 4.1.2

Rewrite $14$ as $6$ plus $8$

$3 x^{2} + (6 + 8) x + 16 = 0$

Step 4.1.3

Apply the distributive property.

$3 x^{2} + 6 x + 8 x + 16 = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$(3 x^{2} + 6 x) + 8 x + 16 = 0$

Step 4.2.2

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

$3 x (x + 2) + 8 (x + 2) = 0$

Step 4.3

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

$(x + 2) (3 x + 8) = 0$

Step 5

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

$x + 2 = 0$

$3 x + 8 = 0$

Step 6

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

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Step 6.1

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

$x + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7

Set $3 x + 8$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $3 x + 8$ equal to $0$ .

$3 x + 8 = 0$

Step 7.2

Solve $3 x + 8 = 0$ for $x$ .

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Step 7.2.1

Subtract $8$ from both sides of the equation.

$3 x = - 8$

Step 7.2.2

Divide each term in $3 x = - 8$ by $3$ and simplify.

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Step 7.2.2.1

Divide each term in $3 x = - 8$ by $3$ .

$\frac{3 x}{3} = \frac{- 8}{3}$

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.

$\frac{3 x}{3} = \frac{- 8}{3}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{3}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 8

The final solution is all the values that make $(x + 2) (3 x + 8) = 0$ true.

$x = - 2, - \frac{8}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 2, - \frac{8}{3}$

Decimal Form:

$x = - 2, - 2.\overline{6}$

Mixed Number Form:

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