Precalculus Examples

${(x + 14)}^{2} + x^{2} = 196$

Step 1

Subtract $196$ from both sides of the equation.

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

Step 2

Simplify ${(x + 14)}^{2} + x^{2} - 196$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 14 + 14 (x + 14) + x^{2} - 196 = 0$

Step 2.1.2.3

Apply the distributive property.

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

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.3.1.2

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

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

Step 2.1.3.1.3

Multiply $14$ by $14$ .

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

Step 2.1.3.2

Add $14 x$ and $14 x$ .

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

Step 2.2

Combine the opposite terms in $x^{2} + 28 x + 196 + x^{2} - 196$ .

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Step 2.2.1

Subtract $196$ from $196$ .

$x^{2} + 28 x + x^{2} + 0 = 0$

Step 2.2.2

Add $x^{2} + 28 x + x^{2}$ and $0$ .

$x^{2} + 28 x + x^{2} = 0$

Step 2.3

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

$2 x^{2} + 28 x = 0$

Step 3

Factor $2 x$ out of $2 x^{2} + 28 x$ .

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Step 3.1

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

$2 x (x) + 28 x = 0$

Step 3.2

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

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

Step 3.3

Factor $2 x$ out of $2 x (x) + 2 x (14)$ .

$2 x (x + 14) = 0$

Step 4

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

$x = 0$

$x + 14 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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Step 6.1

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

$x + 14 = 0$

Step 6.2

Subtract $14$ from both sides of the equation.

$x = - 14$

Step 7

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

$x = 0, - 14$