Algebra Examples

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

Step 1

Simplify $4 {(x - 3)}^{2}$ .

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Step 1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

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

Step 1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 3$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} = 4 (x^{2} - 3 x - 3 x + 9)$

Step 1.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} = 4 (x^{2} - 6 x + 9)$

Step 1.4

Apply the distributive property.

$x^{2} = 4 x^{2} + 4 (- 6 x) + 4 \cdot 9$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 6$ by $4$ .

$x^{2} = 4 x^{2} - 24 x + 4 \cdot 9$

Step 1.5.2

Multiply $4$ by $9$ .

$x^{2} = 4 x^{2} - 24 x + 36$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 24 x + 36 = x^{2}$

Step 3

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

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Step 3.1

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

$4 x^{2} - 24 x + 36 - x^{2} = 0$

Step 3.2

Subtract $x^{2}$ from $4 x^{2}$ .

$3 x^{2} - 24 x + 36 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $3$ out of $3 x^{2} - 24 x + 36$ .

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Step 4.1.1

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

$3 (x^{2}) - 24 x + 36 = 0$

Step 4.1.2

Factor $3$ out of $- 24 x$ .

$3 (x^{2}) + 3 (- 8 x) + 36 = 0$

Step 4.1.3

Factor $3$ out of $36$ .

$3 x^{2} + 3 (- 8 x) + 3 \cdot 12 = 0$

Step 4.1.4

Factor $3$ out of $3 x^{2} + 3 (- 8 x)$ .

$3 (x^{2} - 8 x) + 3 \cdot 12 = 0$

Step 4.1.5

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 12$ .

$3 (x^{2} - 8 x + 12) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} - 8 x + 12$ using the AC method.

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Step 4.2.1.1

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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 4.2.1.2

Write the factored form using these integers.

$3 ((x - 6) (x - 2)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$3 (x - 6) (x - 2) = 0$

Step 5

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

$x - 6 = 0$

$x - 2 = 0$

Step 6

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 6.2

Add $6$ to both sides of the equation.

$x = 6$

Step 7

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

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Step 7.1

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

$x - 2 = 0$

Step 7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 8

The final solution is all the values that make $3 (x - 6) (x - 2) = 0$ true.

$x = 6, 2$