Pre-Algebra Examples

$x^{2} + {(x - 14)}^{2} = 26^{2}$

Step 1

Raise $26$ to the power of $2$ .

$x^{2} + {(x - 14)}^{2} = 676$

Step 2

Subtract $676$ from both sides of the equation.

$x^{2} + {(x - 14)}^{2} - 676 = 0$

Step 3

Simplify $x^{2} + {(x - 14)}^{2} - 676$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$x^{2} + (x - 14) (x - 14) - 676 = 0$

Step 3.1.2

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

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Step 3.1.2.1

Apply the distributive property.

$x^{2} + x (x - 14) - 14 (x - 14) - 676 = 0$

Step 3.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 14 - 14 (x - 14) - 676 = 0$

Step 3.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14 - 676 = 0$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot - 14 - 14 x - 14 \cdot - 14 - 676 = 0$

Step 3.1.3.1.2

Move $- 14$ to the left of $x$ .

$x^{2} + x^{2} - 14 \cdot x - 14 x - 14 \cdot - 14 - 676 = 0$

Step 3.1.3.1.3

Multiply $- 14$ by $- 14$ .

$x^{2} + x^{2} - 14 x - 14 x + 196 - 676 = 0$

Step 3.1.3.2

Subtract $14 x$ from $- 14 x$ .

$x^{2} + x^{2} - 28 x + 196 - 676 = 0$

Step 3.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 28 x + 196 - 676 = 0$

Step 3.3

Subtract $676$ from $196$ .

$2 x^{2} - 28 x - 480 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $2$ out of $2 x^{2} - 28 x - 480$ .

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Step 4.1.1

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

$2 (x^{2}) - 28 x - 480 = 0$

Step 4.1.2

Factor $2$ out of $- 28 x$ .

$2 (x^{2}) + 2 (- 14 x) - 480 = 0$

Step 4.1.3

Factor $2$ out of $- 480$ .

$2 x^{2} + 2 (- 14 x) + 2 \cdot - 240 = 0$

Step 4.1.4

Factor $2$ out of $2 x^{2} + 2 (- 14 x)$ .

$2 (x^{2} - 14 x) + 2 \cdot - 240 = 0$

Step 4.1.5

Factor $2$ out of $2 (x^{2} - 14 x) + 2 \cdot - 240$ .

$2 (x^{2} - 14 x - 240) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} - 14 x - 240$ 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 $- 240$ and whose sum is $- 14$ .

$- 24, 10$

Step 4.2.1.2

Write the factored form using these integers.

$2 ((x - 24) (x + 10)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$2 (x - 24) (x + 10) = 0$

Step 5

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

$x - 24 = 0$

$x + 10 = 0$

Step 6

Set $x - 24$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 24$ equal to $0$ .

$x - 24 = 0$

Step 6.2

Add $24$ to both sides of the equation.

$x = 24$

Step 7

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 7.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 8

The final solution is all the values that make $2 (x - 24) (x + 10) = 0$ true.

$x = 24, - 10$