Precalculus Examples

$x^{2} + 10 x y + y^{2} - 8 = 0$

Step 1

Solve for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 1$ , $b = 10 x$ , and $c = x^{2} - 8$ into the quadratic formula and solve for $y$ .

$\frac{- 10 x \pm \sqrt{{(10 x)}^{2} - 4 \cdot (1 \cdot (x^{2} - 8))}}{2 \cdot 1}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Add parentheses.

$y = \frac{- 10 x \pm \sqrt{{(10 x)}^{2} - 4 \cdot (1 \cdot (x^{2} - 8))}}{2 \cdot 1}$

Step 1.3.1.2

Let $u = 1 \cdot (x^{2} - 8)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8)$ .

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Step 1.3.1.2.1

Apply the product rule to $10 x$ .

$y = \frac{- 10 x \pm \sqrt{10^{2} x^{2} - 4 \cdot u}}{2 \cdot 1}$

Step 1.3.1.2.2

Raise $10$ to the power of $2$ .

$y = \frac{- 10 x \pm \sqrt{100 x^{2} - 4 u}}{2 \cdot 1}$

Step 1.3.1.3

Factor $4$ out of $100 x^{2} - 4 u$ .

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Step 1.3.1.3.1

Factor $4$ out of $100 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2}) - 4 u}}{2 \cdot 1}$

Step 1.3.1.3.2

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

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2}) + 4 (- u)}}{2 \cdot 1}$

Step 1.3.1.3.3

Factor $4$ out of $4 (25 x^{2}) + 4 (- u)$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - u)}}{2 \cdot 1}$

Step 1.3.1.4

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8)$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - (1 \cdot (x^{2} - 8)))}}{2 \cdot 1}$

Step 1.3.1.5

Simplify.

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Step 1.3.1.5.1

Simplify each term.

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Step 1.3.1.5.1.1

Multiply $x^{2} - 8$ by $1$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - (x^{2} - 8))}}{2 \cdot 1}$

Step 1.3.1.5.1.2

Apply the distributive property.

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - x^{2} + 8)}}{2 \cdot 1}$

Step 1.3.1.5.1.3

Multiply $- 1$ by $- 8$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - x^{2} + 8)}}{2 \cdot 1}$

Step 1.3.1.5.2

Subtract $x^{2}$ from $25 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (24 x^{2} + 8)}}{2 \cdot 1}$

Step 1.3.1.6

Factor $8$ out of $24 x^{2} + 8$ .

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Step 1.3.1.6.1

Factor $8$ out of $24 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2}) + 8)}}{2 \cdot 1}$

Step 1.3.1.6.2

Factor $8$ out of $8$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2}) + 8 (1))}}{2 \cdot 1}$

Step 1.3.1.6.3

Factor $8$ out of $8 (3 x^{2}) + 8 (1)$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2} + 1))}}{2 \cdot 1}$

$y = \frac{- 10 x \pm \sqrt{4 \cdot (8 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.3.1.7

Multiply $4$ by $8$ .

$y = \frac{- 10 x \pm \sqrt{32 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.3.1.8

Rewrite $32 (3 x^{2} + 1)$ as ${(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))$ .

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Step 1.3.1.8.1

Factor $16$ out of $32$ .

$y = \frac{- 10 x \pm \sqrt{16 (2) (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.3.1.8.2

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

$y = \frac{- 10 x \pm \sqrt{4^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.3.1.8.3

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

$y = \frac{- 10 x \pm \sqrt{{(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.3.1.8.4

Add parentheses.

$y = \frac{- 10 x \pm \sqrt{{(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{- 10 x \pm 2^{2} \sqrt{2 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.3.1.10

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

$y = \frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.3.2

Multiply $2$ by $1$ .

$y = \frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2}$

Step 1.3.3

Simplify $\frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2}$ .

$y = - 5 x \pm 2 \sqrt{2 (3 x^{2} + 1)}$

Step 1.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Add parentheses.

$y = \frac{- 10 x \pm \sqrt{{(10 x)}^{2} - 4 \cdot (1 \cdot (x^{2} - 8))}}{2 \cdot 1}$

Step 1.4.1.2

Let $u = 1 \cdot (x^{2} - 8)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8)$ .

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Step 1.4.1.2.1

Apply the product rule to $10 x$ .

$y = \frac{- 10 x \pm \sqrt{10^{2} x^{2} - 4 \cdot u}}{2 \cdot 1}$

Step 1.4.1.2.2

Raise $10$ to the power of $2$ .

$y = \frac{- 10 x \pm \sqrt{100 x^{2} - 4 u}}{2 \cdot 1}$

Step 1.4.1.3

Factor $4$ out of $100 x^{2} - 4 u$ .

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Step 1.4.1.3.1

Factor $4$ out of $100 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2}) - 4 u}}{2 \cdot 1}$

Step 1.4.1.3.2

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

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2}) + 4 (- u)}}{2 \cdot 1}$

Step 1.4.1.3.3

Factor $4$ out of $4 (25 x^{2}) + 4 (- u)$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - u)}}{2 \cdot 1}$

Step 1.4.1.4

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8)$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - (1 \cdot (x^{2} - 8)))}}{2 \cdot 1}$

Step 1.4.1.5

Simplify.

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Step 1.4.1.5.1

Simplify each term.

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Step 1.4.1.5.1.1

Multiply $x^{2} - 8$ by $1$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - (x^{2} - 8))}}{2 \cdot 1}$

Step 1.4.1.5.1.2

Apply the distributive property.

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - x^{2} + 8)}}{2 \cdot 1}$

Step 1.4.1.5.1.3

Multiply $- 1$ by $- 8$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - x^{2} + 8)}}{2 \cdot 1}$

Step 1.4.1.5.2

Subtract $x^{2}$ from $25 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (24 x^{2} + 8)}}{2 \cdot 1}$

Step 1.4.1.6

Factor $8$ out of $24 x^{2} + 8$ .

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Step 1.4.1.6.1

Factor $8$ out of $24 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2}) + 8)}}{2 \cdot 1}$

Step 1.4.1.6.2

Factor $8$ out of $8$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2}) + 8 (1))}}{2 \cdot 1}$

Step 1.4.1.6.3

Factor $8$ out of $8 (3 x^{2}) + 8 (1)$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2} + 1))}}{2 \cdot 1}$

$y = \frac{- 10 x \pm \sqrt{4 \cdot (8 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.4.1.7

Multiply $4$ by $8$ .

$y = \frac{- 10 x \pm \sqrt{32 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.4.1.8

Rewrite $32 (3 x^{2} + 1)$ as ${(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))$ .

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Step 1.4.1.8.1

Factor $16$ out of $32$ .

$y = \frac{- 10 x \pm \sqrt{16 (2) (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.4.1.8.2

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

$y = \frac{- 10 x \pm \sqrt{4^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.4.1.8.3

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

$y = \frac{- 10 x \pm \sqrt{{(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.4.1.8.4

Add parentheses.

$y = \frac{- 10 x \pm \sqrt{{(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{- 10 x \pm 2^{2} \sqrt{2 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.4.1.10

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

$y = \frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2}$

Step 1.4.3

Simplify $\frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2}$ .

$y = - 5 x \pm 2 \sqrt{2 (3 x^{2} + 1)}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = - 5 x + 2 \sqrt{2 (3 x^{2} + 1)}$

Step 1.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Add parentheses.

$y = \frac{- 10 x \pm \sqrt{{(10 x)}^{2} - 4 \cdot (1 \cdot (x^{2} - 8))}}{2 \cdot 1}$

Step 1.5.1.2

Let $u = 1 \cdot (x^{2} - 8)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8)$ .

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Step 1.5.1.2.1

Apply the product rule to $10 x$ .

$y = \frac{- 10 x \pm \sqrt{10^{2} x^{2} - 4 \cdot u}}{2 \cdot 1}$

Step 1.5.1.2.2

Raise $10$ to the power of $2$ .

$y = \frac{- 10 x \pm \sqrt{100 x^{2} - 4 u}}{2 \cdot 1}$

Step 1.5.1.3

Factor $4$ out of $100 x^{2} - 4 u$ .

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Step 1.5.1.3.1

Factor $4$ out of $100 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2}) - 4 u}}{2 \cdot 1}$

Step 1.5.1.3.2

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

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2}) + 4 (- u)}}{2 \cdot 1}$

Step 1.5.1.3.3

Factor $4$ out of $4 (25 x^{2}) + 4 (- u)$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - u)}}{2 \cdot 1}$

Step 1.5.1.4

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8)$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - (1 \cdot (x^{2} - 8)))}}{2 \cdot 1}$

Step 1.5.1.5

Simplify.

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Step 1.5.1.5.1

Simplify each term.

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Step 1.5.1.5.1.1

Multiply $x^{2} - 8$ by $1$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - (x^{2} - 8))}}{2 \cdot 1}$

Step 1.5.1.5.1.2

Apply the distributive property.

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - x^{2} + 8)}}{2 \cdot 1}$

Step 1.5.1.5.1.3

Multiply $- 1$ by $- 8$ .

$y = \frac{- 10 x \pm \sqrt{4 (25 x^{2} - x^{2} + 8)}}{2 \cdot 1}$

Step 1.5.1.5.2

Subtract $x^{2}$ from $25 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (24 x^{2} + 8)}}{2 \cdot 1}$

Step 1.5.1.6

Factor $8$ out of $24 x^{2} + 8$ .

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Step 1.5.1.6.1

Factor $8$ out of $24 x^{2}$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2}) + 8)}}{2 \cdot 1}$

Step 1.5.1.6.2

Factor $8$ out of $8$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2}) + 8 (1))}}{2 \cdot 1}$

Step 1.5.1.6.3

Factor $8$ out of $8 (3 x^{2}) + 8 (1)$ .

$y = \frac{- 10 x \pm \sqrt{4 (8 (3 x^{2} + 1))}}{2 \cdot 1}$

$y = \frac{- 10 x \pm \sqrt{4 \cdot (8 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.5.1.7

Multiply $4$ by $8$ .

$y = \frac{- 10 x \pm \sqrt{32 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.5.1.8

Rewrite $32 (3 x^{2} + 1)$ as ${(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))$ .

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Step 1.5.1.8.1

Factor $16$ out of $32$ .

$y = \frac{- 10 x \pm \sqrt{16 (2) (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.5.1.8.2

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

$y = \frac{- 10 x \pm \sqrt{4^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.5.1.8.3

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

$y = \frac{- 10 x \pm \sqrt{{(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.5.1.8.4

Add parentheses.

$y = \frac{- 10 x \pm \sqrt{{(2^{2})}^{2} \cdot (2 (3 x^{2} + 1))}}{2 \cdot 1}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{- 10 x \pm 2^{2} \sqrt{2 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.5.1.10

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

$y = \frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2}$

Step 1.5.3

Simplify $\frac{- 10 x \pm 4 \sqrt{2 (3 x^{2} + 1)}}{2}$ .

$y = - 5 x \pm 2 \sqrt{2 (3 x^{2} + 1)}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = - 5 x - 2 \sqrt{2 (3 x^{2} + 1)}$

Step 1.6

The final answer is the combination of both solutions.

$y = - 5 x + 2 \sqrt{2 (3 x^{2} + 1)}$

$y = - 5 x - 2 \sqrt{2 (3 x^{2} + 1)}$

$y = - 5 x + 2 \sqrt{2 (3 x^{2} + 1)}$

$y = - 5 x - 2 \sqrt{2 (3 x^{2} + 1)}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

The standard form is $- 5 x + 2 \sqrt{2 (3 x^{2} + 1)}, - 5 x - 2 \sqrt{2 (3 x^{2} + 1)}$ .

$y = - 5 x + 2 \sqrt{2 (3 x^{2} + 1)}$

$y = - 5 x - 2 \sqrt{2 (3 x^{2} + 1)}$

Step 4