Pre-Algebra Examples

$8 {(5 x - 1)}^{2} + 2 = - 8 (5 x - 1)$

Step 1

Simplify $- 8 (5 x - 1)$ .

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Step 1.1

Rewrite.

$8 {(5 x - 1)}^{2} + 2 = 0 + 0 - 8 (5 x - 1)$

Step 1.2

Simplify by adding zeros.

$8 {(5 x - 1)}^{2} + 2 = - 8 (5 x - 1)$

Step 1.3

Apply the distributive property.

$8 {(5 x - 1)}^{2} + 2 = - 8 (5 x) - 8 \cdot - 1$

Step 1.4

Multiply.

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Step 1.4.1

Multiply $5$ by $- 8$ .

$8 {(5 x - 1)}^{2} + 2 = - 40 x - 8 \cdot - 1$

Step 1.4.2

Multiply $- 8$ by $- 1$ .

$8 {(5 x - 1)}^{2} + 2 = - 40 x + 8$

Step 2

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

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Step 2.1

Add $40 x$ to both sides of the equation.

$8 {(5 x - 1)}^{2} + 2 + 40 x = 8$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(5 x - 1)}^{2}$ as $(5 x - 1) (5 x - 1)$ .

$8 ((5 x - 1) (5 x - 1)) + 2 + 40 x = 8$

Step 2.2.2

Expand $(5 x - 1) (5 x - 1)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$8 (5 x (5 x - 1) - 1 (5 x - 1)) + 2 + 40 x = 8$

Step 2.2.2.2

Apply the distributive property.

$8 (5 x (5 x) + 5 x \cdot - 1 - 1 (5 x - 1)) + 2 + 40 x = 8$

Step 2.2.2.3

Apply the distributive property.

$8 (5 x (5 x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite using the commutative property of multiplication.

$8 (5 \cdot 5 x \cdot x + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

Move $x$ .

$8 (5 \cdot 5 (x \cdot x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3.1.2.2

Multiply $x$ by $x$ .

$8 (5 \cdot 5 x^{2} + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3.1.3

Multiply $5$ by $5$ .

$8 (25 x^{2} + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3.1.4

Multiply $- 1$ by $5$ .

$8 (25 x^{2} - 5 x - 1 (5 x) - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3.1.5

Multiply $5$ by $- 1$ .

$8 (25 x^{2} - 5 x - 5 x - 1 \cdot - 1) + 2 + 40 x = 8$

Step 2.2.3.1.6

Multiply $- 1$ by $- 1$ .

$8 (25 x^{2} - 5 x - 5 x + 1) + 2 + 40 x = 8$

Step 2.2.3.2

Subtract $5 x$ from $- 5 x$ .

$8 (25 x^{2} - 10 x + 1) + 2 + 40 x = 8$

Step 2.2.4

Apply the distributive property.

$8 (25 x^{2}) + 8 (- 10 x) + 8 \cdot 1 + 2 + 40 x = 8$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Multiply $25$ by $8$ .

$200 x^{2} + 8 (- 10 x) + 8 \cdot 1 + 2 + 40 x = 8$

Step 2.2.5.2

Multiply $- 10$ by $8$ .

$200 x^{2} - 80 x + 8 \cdot 1 + 2 + 40 x = 8$

Step 2.2.5.3

Multiply $8$ by $1$ .

$200 x^{2} - 80 x + 8 + 2 + 40 x = 8$

Step 2.3

Add $- 80 x$ and $40 x$ .

$200 x^{2} - 40 x + 8 + 2 = 8$

Step 2.4

Add $8$ and $2$ .

$200 x^{2} - 40 x + 10 = 8$

Step 3

Subtract $8$ from both sides of the equation.

$200 x^{2} - 40 x + 10 - 8 = 0$

Step 4

Subtract $8$ from $10$ .

$200 x^{2} - 40 x + 2 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $2$ out of $200 x^{2} - 40 x + 2$ .

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Step 5.1.1

Factor $2$ out of $200 x^{2}$ .

$2 (100 x^{2}) - 40 x + 2 = 0$

Step 5.1.2

Factor $2$ out of $- 40 x$ .

$2 (100 x^{2}) + 2 (- 20 x) + 2 = 0$

Step 5.1.3

Factor $2$ out of $2$ .

$2 (100 x^{2}) + 2 (- 20 x) + 2 (1) = 0$

Step 5.1.4

Factor $2$ out of $2 (100 x^{2}) + 2 (- 20 x)$ .

$2 (100 x^{2} - 20 x) + 2 (1) = 0$

Step 5.1.5

Factor $2$ out of $2 (100 x^{2} - 20 x) + 2 (1)$ .

$2 (100 x^{2} - 20 x + 1) = 0$

Step 5.2

Factor using the perfect square rule.

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Step 5.2.1

Rewrite $100 x^{2}$ as ${(10 x)}^{2}$ .

$2 ({(10 x)}^{2} - 20 x + 1) = 0$

Step 5.2.2

Rewrite $1$ as $1^{2}$ .

$2 ({(10 x)}^{2} - 20 x + 1^{2}) = 0$

Step 5.2.3

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

$20 x = 2 \cdot (10 x) \cdot 1$

Step 5.2.4

Rewrite the polynomial.

$2 ({(10 x)}^{2} - 2 \cdot (10 x) \cdot 1 + 1^{2}) = 0$

Step 5.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 10 x$ and $b = 1$ .

$2 {(10 x - 1)}^{2} = 0$

Step 6

Divide each term in $2 {(10 x - 1)}^{2} = 0$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 {(10 x - 1)}^{2} = 0$ by $2$ .

$\frac{2 {(10 x - 1)}^{2}}{2} = \frac{0}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 {(10 x - 1)}^{2}}{2} = \frac{0}{2}$

Step 6.2.1.2

Divide ${(10 x - 1)}^{2}$ by $1$ .

${(10 x - 1)}^{2} = \frac{0}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $2$ .

${(10 x - 1)}^{2} = 0$

Step 7

Set the $10 x - 1$ equal to $0$ .

$10 x - 1 = 0$

Step 8

Solve for $x$ .

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Step 8.1

Add $1$ to both sides of the equation.

$10 x = 1$

Step 8.2

Divide each term in $10 x = 1$ by $10$ and simplify.

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Step 8.2.1

Divide each term in $10 x = 1$ by $10$ .

$\frac{10 x}{10} = \frac{1}{10}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $10$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{1}{10}$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{10}$