Algebra Examples

$3 x (x - 2) - (x - 6) = 23 (x - 3)$

Step 1

Simplify $3 x (x - 2) - (x - 6)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 2 - (x - 6) = 23 (x - 3)$

Step 1.1.2

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

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Step 1.1.2.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 2 - (x - 6) = 23 (x - 3)$

Step 1.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 2 - (x - 6) = 23 (x - 3)$

Step 1.1.3

Multiply $- 2$ by $3$ .

$3 x^{2} - 6 x - (x - 6) = 23 (x - 3)$

Step 1.1.4

Apply the distributive property.

$3 x^{2} - 6 x - x - - 6 = 23 (x - 3)$

Step 1.1.5

Multiply $- 1$ by $- 6$ .

$3 x^{2} - 6 x - x + 6 = 23 (x - 3)$

Step 1.2

Subtract $x$ from $- 6 x$ .

$3 x^{2} - 7 x + 6 = 23 (x - 3)$

Step 2

Simplify $23 (x - 3)$ .

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Step 2.1

Apply the distributive property.

$3 x^{2} - 7 x + 6 = 23 x + 23 \cdot - 3$

Step 2.2

Multiply $23$ by $- 3$ .

$3 x^{2} - 7 x + 6 = 23 x - 69$

Step 3

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

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Step 3.1

Subtract $23 x$ from both sides of the equation.

$3 x^{2} - 7 x + 6 - 23 x = - 69$

Step 3.2

Subtract $23 x$ from $- 7 x$ .

$3 x^{2} - 30 x + 6 = - 69$

Step 4

Add $69$ to both sides of the equation.

$3 x^{2} - 30 x + 6 + 69 = 0$

Step 5

Add $6$ and $69$ .

$3 x^{2} - 30 x + 75 = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $3$ out of $3 x^{2} - 30 x + 75$ .

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Step 6.1.1

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

$3 (x^{2}) - 30 x + 75 = 0$

Step 6.1.2

Factor $3$ out of $- 30 x$ .

$3 (x^{2}) + 3 (- 10 x) + 75 = 0$

Step 6.1.3

Factor $3$ out of $75$ .

$3 x^{2} + 3 (- 10 x) + 3 \cdot 25 = 0$

Step 6.1.4

Factor $3$ out of $3 x^{2} + 3 (- 10 x)$ .

$3 (x^{2} - 10 x) + 3 \cdot 25 = 0$

Step 6.1.5

Factor $3$ out of $3 (x^{2} - 10 x) + 3 \cdot 25$ .

$3 (x^{2} - 10 x + 25) = 0$

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

Rewrite $25$ as $5^{2}$ .

$3 (x^{2} - 10 x + 5^{2}) = 0$

Step 6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 6.2.3

Rewrite the polynomial.

$3 (x^{2} - 2 \cdot x \cdot 5 + 5^{2}) = 0$

Step 6.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$3 {(x - 5)}^{2} = 0$

Step 7

Divide each term in $3 {(x - 5)}^{2} = 0$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 {(x - 5)}^{2} = 0$ by $3$ .

$\frac{3 {(x - 5)}^{2}}{3} = \frac{0}{3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 {(x - 5)}^{2}}{3} = \frac{0}{3}$

Step 7.2.1.2

Divide ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} = \frac{0}{3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $3$ .

${(x - 5)}^{2} = 0$

Step 8

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 9

Add $5$ to both sides of the equation.

$x = 5$