Pre-Algebra Examples

$x^{2} + {(\frac{4 x + 25}{3})}^{2} = 25$

Step 1

Simplify $x^{2} + {(\frac{4 x + 25}{3})}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the product rule to $\frac{4 x + 25}{3}$ .

$x^{2} + \frac{{(4 x + 25)}^{2}}{3^{2}} = 25$

Step 1.1.2

Raise $3$ to the power of $2$ .

$x^{2} + \frac{{(4 x + 25)}^{2}}{9} = 25$

Step 1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x^{2} \cdot \frac{9}{9} + \frac{{(4 x + 25)}^{2}}{9} = 25$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $x^{2}$ and $\frac{9}{9}$ .

$\frac{x^{2} \cdot 9}{9} + \frac{{(4 x + 25)}^{2}}{9} = 25$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 9 + {(4 x + 25)}^{2}}{9} = 25$

Step 1.4

Move $9$ to the left of $x^{2}$ .

$\frac{9 x^{2} + {(4 x + 25)}^{2}}{9} = 25$

Step 2

Multiply both sides by $9$ .

$\frac{9 x^{2} + {(4 x + 25)}^{2}}{9} \cdot 9 = 25 \cdot 9$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $9$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{9 x^{2} + {(4 x + 25)}^{2}}{9} \cdot 9 = 25 \cdot 9$

Step 3.1.1.2

Rewrite the expression.

$9 x^{2} + {(4 x + 25)}^{2} = 25 \cdot 9$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $25$ by $9$ .

$9 x^{2} + {(4 x + 25)}^{2} = 225$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $225$ from both sides of the equation.

$9 x^{2} + {(4 x + 25)}^{2} - 225 = 0$

Step 4.2

Simplify $9 x^{2} + {(4 x + 25)}^{2} - 225$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite ${(4 x + 25)}^{2}$ as $(4 x + 25) (4 x + 25)$ .

$9 x^{2} + (4 x + 25) (4 x + 25) - 225 = 0$

Step 4.2.1.2

Expand $(4 x + 25) (4 x + 25)$ using the FOIL Method.

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Step 4.2.1.2.1

Apply the distributive property.

$9 x^{2} + 4 x (4 x + 25) + 25 (4 x + 25) - 225 = 0$

Step 4.2.1.2.2

Apply the distributive property.

$9 x^{2} + 4 x (4 x) + 4 x \cdot 25 + 25 (4 x + 25) - 225 = 0$

Step 4.2.1.2.3

Apply the distributive property.

$9 x^{2} + 4 x (4 x) + 4 x \cdot 25 + 25 (4 x) + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3

Simplify and combine like terms.

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Step 4.2.1.3.1

Simplify each term.

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Step 4.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$9 x^{2} + 4 \cdot 4 x \cdot x + 4 x \cdot 25 + 25 (4 x) + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3.1.2

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

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Step 4.2.1.3.1.2.1

Move $x$ .

$9 x^{2} + 4 \cdot 4 (x \cdot x) + 4 x \cdot 25 + 25 (4 x) + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3.1.2.2

Multiply $x$ by $x$ .

$9 x^{2} + 4 \cdot 4 x^{2} + 4 x \cdot 25 + 25 (4 x) + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3.1.3

Multiply $4$ by $4$ .

$9 x^{2} + 16 x^{2} + 4 x \cdot 25 + 25 (4 x) + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3.1.4

Multiply $25$ by $4$ .

$9 x^{2} + 16 x^{2} + 100 x + 25 (4 x) + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3.1.5

Multiply $4$ by $25$ .

$9 x^{2} + 16 x^{2} + 100 x + 100 x + 25 \cdot 25 - 225 = 0$

Step 4.2.1.3.1.6

Multiply $25$ by $25$ .

$9 x^{2} + 16 x^{2} + 100 x + 100 x + 625 - 225 = 0$

Step 4.2.1.3.2

Add $100 x$ and $100 x$ .

$9 x^{2} + 16 x^{2} + 200 x + 625 - 225 = 0$

Step 4.2.2

Add $9 x^{2}$ and $16 x^{2}$ .

$25 x^{2} + 200 x + 625 - 225 = 0$

Step 4.2.3

Subtract $225$ from $625$ .

$25 x^{2} + 200 x + 400 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $25$ out of $25 x^{2} + 200 x + 400$ .

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Step 4.3.1.1

Factor $25$ out of $25 x^{2}$ .

$25 (x^{2}) + 200 x + 400 = 0$

Step 4.3.1.2

Factor $25$ out of $200 x$ .

$25 (x^{2}) + 25 (8 x) + 400 = 0$

Step 4.3.1.3

Factor $25$ out of $400$ .

$25 x^{2} + 25 (8 x) + 25 \cdot 16 = 0$

Step 4.3.1.4

Factor $25$ out of $25 x^{2} + 25 (8 x)$ .

$25 (x^{2} + 8 x) + 25 \cdot 16 = 0$

Step 4.3.1.5

Factor $25$ out of $25 (x^{2} + 8 x) + 25 \cdot 16$ .

$25 (x^{2} + 8 x + 16) = 0$

Step 4.3.2

Factor using the perfect square rule.

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Step 4.3.2.1

Rewrite $16$ as $4^{2}$ .

$25 (x^{2} + 8 x + 4^{2}) = 0$

Step 4.3.2.2

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

$8 x = 2 \cdot x \cdot 4$

Step 4.3.2.3

Rewrite the polynomial.

$25 (x^{2} + 2 \cdot x \cdot 4 + 4^{2}) = 0$

Step 4.3.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$25 {(x + 4)}^{2} = 0$

Step 4.4

Divide each term in $25 {(x + 4)}^{2} = 0$ by $25$ and simplify.

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Step 4.4.1

Divide each term in $25 {(x + 4)}^{2} = 0$ by $25$ .

$\frac{25 {(x + 4)}^{2}}{25} = \frac{0}{25}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $25$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{25 {(x + 4)}^{2}}{25} = \frac{0}{25}$

Step 4.4.2.1.2

Divide ${(x + 4)}^{2}$ by $1$ .

${(x + 4)}^{2} = \frac{0}{25}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $25$ .

${(x + 4)}^{2} = 0$

Step 4.5

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.6

Subtract $4$ from both sides of the equation.

$x = - 4$