Precalculus Examples

$0.01 x^{2} - 2.06 x y + 157.65 y^{2} + 5408.39 x + 24.8 y = - 24751.67$

Step 1

Add $24751.67$ to both sides of the equation.

$0.01 x^{2} - 2.06 x y + 157.65 y^{2} + 5408.39 x + 24.8 y + 24751.67 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 0.01$ , $b = - 2.06 y + 5408.39$ , and $c = 157.65 y^{2} + 24.8 y + 24751.67$ into the quadratic formula and solve for $x$ .

$\frac{- (- 2.06 y + 5408.39) \pm \sqrt{{(- 2.06 y + 5408.39)}^{2} - 4 \cdot (0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67))}}{2 \cdot 0.01}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$x = \frac{- (- 2.06 y) - 1 \cdot 5408.39 \pm \sqrt{{(- 2.06 y + 5408.39)}^{2} - 4 \cdot 0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)}}{2 \cdot 0.01}$

Step 4.1.2

Multiply $- 2.06$ by $- 1$ .

$x = \frac{2.06 y - 1 \cdot 5408.39 \pm \sqrt{{(- 2.06 y + 5408.39)}^{2} - 4 \cdot 0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)}}{2 \cdot 0.01}$

Step 4.1.3

Multiply $- 1$ by $5408.39$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{{(- 2.06 y + 5408.39)}^{2} - 4 \cdot 0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)}}{2 \cdot 0.01}$

Step 4.1.4

Add parentheses.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{{(- 2.06 y + 5408.39)}^{2} - 4 \cdot (0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67))}}{2 \cdot 0.01}$

Step 4.1.5

Let $u = 0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)$ . Substitute $u$ for all occurrences of $0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)$ .

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Step 4.1.5.1

Rewrite ${(- 2.06 y + 5408.39)}^{2}$ as $(- 2.06 y + 5408.39) (- 2.06 y + 5408.39)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{(- 2.06 y + 5408.39) (- 2.06 y + 5408.39) - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.2

Expand $(- 2.06 y + 5408.39) (- 2.06 y + 5408.39)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{- 2.06 y (- 2.06 y + 5408.39) + 5408.39 (- 2.06 y + 5408.39) - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.2.2

Apply the distributive property.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{- 2.06 y (- 2.06 y) - 2.06 y \cdot 5408.39 + 5408.39 (- 2.06 y + 5408.39) - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.2.3

Apply the distributive property.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{- 2.06 y (- 2.06 y) - 2.06 y \cdot 5408.39 + 5408.39 (- 2.06 y) + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3

Simplify and combine like terms.

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Step 4.1.5.3.1

Simplify each term.

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Step 4.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{- 2.06 \cdot (- 2.06 y \cdot y) - 2.06 y \cdot 5408.39 + 5408.39 (- 2.06 y) + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.5.3.1.2.1

Move $y$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{- 2.06 \cdot (- 2.06 (y \cdot y)) - 2.06 y \cdot 5408.39 + 5408.39 (- 2.06 y) + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.1.2.2

Multiply $y$ by $y$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{- 2.06 \cdot (- 2.06 y^{2}) - 2.06 y \cdot 5408.39 + 5408.39 (- 2.06 y) + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.1.3

Multiply $- 2.06$ by $- 2.06$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{4.2436 y^{2} - 2.06 y \cdot 5408.39 + 5408.39 (- 2.06 y) + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.1.4

Multiply $5408.39$ by $- 2.06$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{4.2436 y^{2} - 11141.2834 y + 5408.39 (- 2.06 y) + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.1.5

Multiply $- 2.06$ by $5408.39$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{4.2436 y^{2} - 11141.2834 y - 11141.2834 y + 5408.39 \cdot 5408.39 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.1.6

Multiply $5408.39$ by $5408.39$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{4.2436 y^{2} - 11141.2834 y - 11141.2834 y + 29250682.3921 - 4 \cdot u}}{2 \cdot 0.01}$

Step 4.1.5.3.2

Subtract $11141.2834 y$ from $- 11141.2834 y$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{4.2436 y^{2} - 22282.5668 y + 29250682.3921 - 4 \cdot u}}{2 \cdot 0.01}$

$x = \frac{2.06 y - 5408.39 \pm \sqrt{4.2436 y^{2} - 22282.5668 y + 29250682.3921 - 4 u}}{2 \cdot 0.01}$

Step 4.1.6

Factor $0.0001$ out of $4.2436 y^{2} - 22282.5668 y + 29250682.3921 - 4 u$ .

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Step 4.1.6.1

Factor $0.0001$ out of $4.2436 y^{2}$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2}) - 22282.5668 y + 29250682.3921 - 4 u}}{2 \cdot 0.01}$

Step 4.1.6.2

Factor $0.0001$ out of $- 22282.5668 y$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2}) + 0.0001 (- 222825668 y) + 29250682.3921 - 4 u}}{2 \cdot 0.01}$

Step 4.1.6.3

Factor $0.0001$ out of $29250682.3921$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2}) + 0.0001 (- 222825668 y) + 0.0001 (292506823921) - 4 u}}{2 \cdot 0.01}$

Step 4.1.6.4

Factor $0.0001$ out of $- 4 u$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2}) + 0.0001 (- 222825668 y) + 0.0001 (292506823921) + 0.0001 (- 40000 u)}}{2 \cdot 0.01}$

Step 4.1.6.5

Factor $0.0001$ out of $0.0001 (42436 y^{2}) + 0.0001 (- 222825668 y)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y) + 0.0001 (292506823921) + 0.0001 (- 40000 u)}}{2 \cdot 0.01}$

Step 4.1.6.6

Factor $0.0001$ out of $0.0001 (42436 y^{2} - 222825668 y) + 0.0001 (292506823921)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921) + 0.0001 (- 40000 u)}}{2 \cdot 0.01}$

Step 4.1.6.7

Factor $0.0001$ out of $0.0001 (42436 y^{2} - 222825668 y + 292506823921) + 0.0001 (- 40000 u)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 u)}}{2 \cdot 0.01}$

Step 4.1.7

Replace all occurrences of $u$ with $0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 (0.01 \cdot (157.65 y^{2} + 24.8 y + 24751.67)))}}{2 \cdot 0.01}$

Step 4.1.8

Simplify.

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Step 4.1.8.1

Simplify each term.

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Step 4.1.8.1.1

Apply the distributive property.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 (0.01 (157.65 y^{2}) + 0.01 (24.8 y) + 0.01 \cdot 24751.67))}}{2 \cdot 0.01}$

Step 4.1.8.1.2

Simplify.

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Step 4.1.8.1.2.1

Multiply $157.65$ by $0.01$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 (1.5765 y^{2} + 0.01 (24.8 y) + 0.01 \cdot 24751.67))}}{2 \cdot 0.01}$

Step 4.1.8.1.2.2

Multiply $24.8$ by $0.01$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 (1.5765 y^{2} + 0.248 y + 0.01 \cdot 24751.67))}}{2 \cdot 0.01}$

Step 4.1.8.1.2.3

Multiply $0.01$ by $24751.67$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 (1.5765 y^{2} + 0.248 y + 247.5167))}}{2 \cdot 0.01}$

Step 4.1.8.1.3

Apply the distributive property.

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 40000 (1.5765 y^{2}) - 40000 (0.248 y) - 40000 \cdot 247.5167)}}{2 \cdot 0.01}$

Step 4.1.8.1.4

Simplify.

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Step 4.1.8.1.4.1

Multiply $1.5765$ by $- 40000$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 63060 y^{2} - 40000 (0.248 y) - 40000 \cdot 247.5167)}}{2 \cdot 0.01}$

Step 4.1.8.1.4.2

Multiply $0.248$ by $- 40000$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 63060 y^{2} - 9920 y - 40000 \cdot 247.5167)}}{2 \cdot 0.01}$

Step 4.1.8.1.4.3

Multiply $- 40000$ by $247.5167$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (42436 y^{2} - 222825668 y + 292506823921 - 63060 y^{2} - 9920 y - 9900668)}}{2 \cdot 0.01}$

Step 4.1.8.2

Subtract $63060 y^{2}$ from $42436 y^{2}$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (- 20624 y^{2} - 222825668 y + 292506823921 - 9920 y - 9900668)}}{2 \cdot 0.01}$

Step 4.1.8.3

Subtract $9920 y$ from $- 222825668 y$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (- 20624 y^{2} - 222835588 y + 292506823921 - 9900668)}}{2 \cdot 0.01}$

Step 4.1.8.4

Subtract $9900668$ from $292506823921$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (- 20624 y^{2} - 222835588 y + 292496923253)}}{2 \cdot 0.01}$

Step 4.1.9

Factor $1$ out of $- 20624 y^{2} - 222835588 y + 292496923253$ .

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Step 4.1.9.1

Factor $1$ out of $- 20624 y^{2}$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (1 (- 20624 y^{2}) - 222835588 y + 292496923253)}}{2 \cdot 0.01}$

Step 4.1.9.2

Factor $1$ out of $- 222835588 y$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (1 (- 20624 y^{2}) + 1 (- 222835588 y) + 292496923253)}}{2 \cdot 0.01}$

Step 4.1.9.3

Rewrite $292496923253$ as $1 (292496923253)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (1 (- 20624 y^{2}) + 1 (- 222835588 y) + 1 (292496923253))}}{2 \cdot 0.01}$

Step 4.1.9.4

Factor $1$ out of $1 (- 20624 y^{2}) + 1 (- 222835588 y)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (1 (- 20624 y^{2} - 222835588 y) + 1 (292496923253))}}{2 \cdot 0.01}$

Step 4.1.9.5

Factor $1$ out of $1 (- 20624 y^{2} - 222835588 y) + 1 (292496923253)$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (1 (- 20624 y^{2} - 222835588 y + 292496923253))}}{2 \cdot 0.01}$

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 \cdot (1 (- 20624 y^{2} - 222835588 y + 292496923253))}}{2 \cdot 0.01}$

Step 4.1.10

Multiply $0.0001$ by $1$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{0.0001 (- 20624 y^{2} - 222835588 y + 292496923253)}}{2 \cdot 0.01}$

Step 4.1.11

Rewrite $0.0001 (- 20624 y^{2} - 222835588 y + 292496923253)$ as ${({0.1}^{2})}^{2} (- 20624 y^{2} - 222835588 y + 292496923253)$ .

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Step 4.1.11.1

Rewrite $0.0001$ as ${0.01}^{2}$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{{0.01}^{2} (- 20624 y^{2} - 222835588 y + 292496923253)}}{2 \cdot 0.01}$

Step 4.1.11.2

Rewrite $0.01$ as ${0.1}^{2}$ .

$x = \frac{2.06 y - 5408.39 \pm \sqrt{{({0.1}^{2})}^{2} (- 20624 y^{2} - 222835588 y + 292496923253)}}{2 \cdot 0.01}$

Step 4.1.12

Pull terms out from under the radical.

$x = \frac{2.06 y - 5408.39 \pm {0.1}^{2} \sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{2 \cdot 0.01}$

Step 4.1.13

Raise $0.1$ to the power of $2$ .

$x = \frac{2.06 y - 5408.39 \pm 0.01 \sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{2 \cdot 0.01}$

Step 4.2

Multiply $2$ by $0.01$ .

$x = \frac{2.06 y - 5408.39 \pm 0.01 \sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{0.02}$

Step 4.3

Simplify $\frac{2.06 y - 5408.39 \pm 0.01 \sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{0.02}$ .

$x = \frac{206 y - 540839}{2} \pm \frac{\sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{2}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{206 y - 540839 + \sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{2}$

$x = \frac{206 y - 540839 - \sqrt{- 20624 y^{2} - 222835588 y + 292496923253}}{2}$