Pre-Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $16 x^{2}$ from both sides of the equation.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(3 x - 14)}^{2}$ as $(3 x - 14) (3 x - 14)$ .

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

Step 1.2.2

Expand $(3 x - 14) (3 x - 14)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Apply the distributive property.

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

Step 1.2.2.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

Move $x$ .

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

Step 1.2.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.2.3.1.3

Multiply $3$ by $3$ .

$9 x^{2} + 3 x \cdot - 14 - 14 (3 x) - 14 \cdot - 14 - 16 x^{2} = 0$

Step 1.2.3.1.4

Multiply $- 14$ by $3$ .

$9 x^{2} - 42 x - 14 (3 x) - 14 \cdot - 14 - 16 x^{2} = 0$

Step 1.2.3.1.5

Multiply $3$ by $- 14$ .

$9 x^{2} - 42 x - 42 x - 14 \cdot - 14 - 16 x^{2} = 0$

Step 1.2.3.1.6

Multiply $- 14$ by $- 14$ .

$9 x^{2} - 42 x - 42 x + 196 - 16 x^{2} = 0$

Step 1.2.3.2

Subtract $42 x$ from $- 42 x$ .

$9 x^{2} - 84 x + 196 - 16 x^{2} = 0$

Step 1.3

Subtract $16 x^{2}$ from $9 x^{2}$ .

$- 7 x^{2} - 84 x + 196 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $- 7$ out of $- 7 x^{2} - 84 x + 196$ .

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Step 2.1.1

Factor $- 7$ out of $- 7 x^{2}$ .

$- 7 x^{2} - 84 x + 196 = 0$

Step 2.1.2

Factor $- 7$ out of $- 84 x$ .

$- 7 x^{2} - 7 (12 x) + 196 = 0$

Step 2.1.3

Factor $- 7$ out of $196$ .

$- 7 x^{2} - 7 (12 x) - 7 \cdot - 28 = 0$

Step 2.1.4

Factor $- 7$ out of $- 7 (x^{2}) - 7 (12 x)$ .

$- 7 (x^{2} + 12 x) - 7 \cdot - 28 = 0$

Step 2.1.5

Factor $- 7$ out of $- 7 (x^{2} + 12 x) - 7 (- 28)$ .

$- 7 (x^{2} + 12 x - 28) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{2} + 12 x - 28$ using the AC method.

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Step 2.2.1.1

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 $- 28$ and whose sum is $12$ .

$- 2, 14$

Step 2.2.1.2

Write the factored form using these integers.

$- 7 ((x - 2) (x + 14)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$- 7 (x - 2) (x + 14) = 0$

Step 3

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

$x - 2 = 0$

$x + 14 = 0$

Step 4

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

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Step 4.1

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

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Set $x + 14$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 14$ equal to $0$ .

$x + 14 = 0$

Step 5.2

Subtract $14$ from both sides of the equation.

$x = - 14$

Step 6

The final solution is all the values that make $- 7 (x - 2) (x + 14) = 0$ true.

$x = 2, - 14$