Solve 7/r - 5 = 3r/ r^2 - 2r - 15

Question

Solve the equation

r_{1} = - \frac{5 + 17}{2}, r_{2} = \frac{- 5 + 17}{2}

Alternative Form

r_{1} \approx - 4.561553, r_{2} \approx - 0.438447

Evaluate

\frac{7}{r} - 5 = (3 \times \frac{r}{r ^{2}}) - 2 r - 15

Find the domain

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\frac{7}{r} - 5 = (3 \times \frac{r}{r ^{2}}) - 2 r - 15, r \neq = 0

Simplify

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\frac{7}{r} - 5 = \frac{3}{r} - 2 r - 15

Multiply both sides of the equation by LCD

(\frac{7}{r} - 5) r = (\frac{3}{r} - 2 r - 15) r

Simplify the equation

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7 - 5 r = (\frac{3}{r} - 2 r - 15) r

Simplify the equation

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7 - 5 r = 3 - 2 r^{2} - 15 r

Move the expression to the left side

7 - 5 r - (3 - 2 r^{2} - 15 r) = 0

Subtract the terms

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4 + 10 r + 2 r^{2} = 0

Rewrite in standard form

2 r^{2} + 10 r + 4 = 0

Substitute a= 2,b= 10 and c= 4 into the quadratic formula r = \frac{- b \pm b ^{2} - 4 a c}{2 a}

r = \frac{- 10 \pm 1 0 ^{2} - 4 \times 2 \times 4}{2 \times 2}

Simplify the expression

r = \frac{- 10 \pm 1 0 ^{2} - 4 \times 2 \times 4}{4}

Simplify the expression

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r = \frac{- 10 \pm 68}{4}

Simplify the radical expression

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r = \frac{- 10 \pm 2 17}{4}

Separate the equation into 2 possible cases

r = \frac{- 10 + 2 17}{4} r = \frac{- 10 - 2 17}{4}

Simplify the expression

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r = \frac{- 5 + 17}{2} r = \frac{- 10 - 2 17}{4}

Simplify the expression

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r = \frac{- 5 + 17}{2} r = - \frac{5 + 17}{2}

Check if the solution is in the defined range

r = \frac{- 5 + 17}{2} r = - \frac{5 + 17}{2}, r \neq = 0

Find the intersection of the solution and the defined range

r = \frac{- 5 + 17}{2} r = - \frac{5 + 17}{2}

Solution

r_{1} = - \frac{5 + 17}{2}, r_{2} = \frac{- 5 + 17}{2}

Alternative Form

r_{1} \approx - 4.561553, r_{2} \approx - 0.438447

Show Solution

Rewrite the equation

21 x^{2} + 21 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2}

Evaluate

\frac{7}{r} - 5 = (3 \times \frac{r}{r ^{2}}) - 2 r - 15

Evaluate

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\frac{7}{r} - 5 = \frac{3}{r} - 2 r - 15

Multiply both sides of the equation by LCD

(\frac{7}{r} - 5) r = (\frac{3}{r} - 2 r - 15) r

Simplify the equation

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7 - 5 r = (\frac{3}{r} - 2 r - 15) r

Simplify the equation

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7 - 5 r = 3 - 2 r^{2} - 15 r

Rewrite the expression

10 r + 2 r^{2} = - 7 + 3

Simplify the expression

10 r + 2 r^{2} = - 4

Use substitution

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10 r + 2 x^{2} + 2 y^{2} = - 4

Simplify the expression

10 r = - 2 x^{2} - 2 y^{2} - 4

Square both sides of the equation

(10 r)^{2} = (- 2 x^{2} - 2 y^{2} - 4)^{2}

Evaluate

100 r^{2} = (- 2 x^{2} - 2 y^{2} - 4)^{2}

To covert the equation to rectangular coordinates using conversion formulas,substitute x^{2} + y^{2} for r^{2}

100 (x^{2} + y^{2}) = (- 2 x^{2} - 2 y^{2} - 4)^{2}

Evaluate the power

100 (x^{2} + y^{2}) = (2 x^{2} + 2 y^{2} + 4)^{2}

Divide both sides of the equation by 4

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

Calculate

25 x^{2} + 25 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2} + 4 x^{2} + 4 y^{2}

Move the expression to the left side

25 x^{2} + 25 y^{2} - (4 x^{2} + 4 y^{2}) = x^{4} + y^{4} + 4 + 2 x^{2} y^{2}

Calculate

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21 x^{2} + 25 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2} + 4 y^{2}

Solution

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21 x^{2} + 21 y^{2} = x^{4} + y^{4} + 4 + 2 x^{2} y^{2}

Show Solution

Solve the equation

Rewrite the equation

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