Finite Math Examples

$\sqrt{x^{2} + 2 x y + y^{2}} = 25$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} + 2 x y + y^{2}}}^{2} = 25^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 2 x y + y^{2}}$ as ${(x^{2} + 2 x y + y^{2})}^{\frac{1}{2}}$ .

${({(x^{2} + 2 x y + y^{2})}^{\frac{1}{2}})}^{2} = 25^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} + 2 x y + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} + 2 x y + y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 2 x y + y^{2})}^{\frac{1}{2} \cdot 2} = 25^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 2 x y + y^{2})}^{\frac{1}{2} \cdot 2} = 25^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 2 x y + y^{2})}^{1} = 25^{2}$

Step 2.2.1.2

Simplify.

$x^{2} + 2 x y + y^{2} = 25^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $25$ to the power of $2$ .

$x^{2} + 2 x y + y^{2} = 625$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $625$ from both sides of the equation.

$x^{2} + 2 x y + y^{2} - 625 = 0$

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3

Substitute the values $a = 1$ , $b = 2 y$ , and $c = y^{2} - 625$ into the quadratic formula and solve for $x$ .

$\frac{- 2 y \pm \sqrt{{(2 y)}^{2} - 4 \cdot (1 \cdot (y^{2} - 625))}}{2 \cdot 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Add parentheses.

$x = \frac{- 2 y \pm \sqrt{{(2 y)}^{2} - 4 \cdot (1 \cdot (y^{2} - 625))}}{2 \cdot 1}$

Step 3.4.1.2

Let $u = 1 \cdot (y^{2} - 625)$ . Substitute $u$ for all occurrences of $1 \cdot (y^{2} - 625)$ .

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Step 3.4.1.2.1

Apply the product rule to $2 y$ .

$x = \frac{- 2 y \pm \sqrt{2^{2} y^{2} - 4 \cdot u}}{2 \cdot 1}$

Step 3.4.1.2.2

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

$x = \frac{- 2 y \pm \sqrt{4 y^{2} - 4 u}}{2 \cdot 1}$

Step 3.4.1.3

Factor $4$ out of $4 y^{2} - 4 u$ .

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Step 3.4.1.3.1

Factor $4$ out of $4 y^{2}$ .

$x = \frac{- 2 y \pm \sqrt{4 (y^{2}) - 4 u}}{2 \cdot 1}$

Step 3.4.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{- 2 y \pm \sqrt{4 (y^{2}) + 4 (- u)}}{2 \cdot 1}$

Step 3.4.1.3.3

Factor $4$ out of $4 (y^{2}) + 4 (- u)$ .

$x = \frac{- 2 y \pm \sqrt{4 (y^{2} - u)}}{2 \cdot 1}$

Step 3.4.1.4

Replace all occurrences of $u$ with $1 \cdot (y^{2} - 625)$ .

$x = \frac{- 2 y \pm \sqrt{4 (y^{2} - (1 \cdot (y^{2} - 625)))}}{2 \cdot 1}$

Step 3.4.1.5

Simplify.

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Step 3.4.1.5.1

Simplify each term.

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Step 3.4.1.5.1.1

Multiply $y^{2} - 625$ by $1$ .

$x = \frac{- 2 y \pm \sqrt{4 (y^{2} - (y^{2} - 625))}}{2 \cdot 1}$

Step 3.4.1.5.1.2

Apply the distributive property.

$x = \frac{- 2 y \pm \sqrt{4 (y^{2} - y^{2} + 625)}}{2 \cdot 1}$

Step 3.4.1.5.1.3

Multiply $- 1$ by $- 625$ .

$x = \frac{- 2 y \pm \sqrt{4 (y^{2} - y^{2} + 625)}}{2 \cdot 1}$

Step 3.4.1.5.2

Subtract $y^{2}$ from $y^{2}$ .

$x = \frac{- 2 y \pm \sqrt{4 (0 + 625)}}{2 \cdot 1}$

Step 3.4.1.5.3

Add $0$ and $625$ .

$x = \frac{- 2 y \pm \sqrt{4 \cdot 625}}{2 \cdot 1}$

Step 3.4.1.6

Multiply $4$ by $625$ .

$x = \frac{- 2 y \pm \sqrt{2500}}{2 \cdot 1}$

Step 3.4.1.7

Rewrite $2500$ as $50^{2}$ .

$x = \frac{- 2 y \pm \sqrt{50^{2}}}{2 \cdot 1}$

Step 3.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 2 y \pm 50}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 y \pm 50}{2}$

Step 3.4.3

Simplify $\frac{- 2 y \pm 50}{2}$ .

$x = - y \pm 25$

Step 3.5

The final answer is the combination of both solutions.

$x = - y + 25$

$x = - y - 25$

$x = - y + 25$

$x = - y - 25$