Solve 11r+15=-2r^2

Question

Solve the quadratic equation

Solve by factoring

Solve using the quadratic formula

Solve by completing the square

r_{1} = - 3, r_{2} = - \frac{5}{2}

Alternative Form

r_{1} = - 3, r_{2} = - 2.5

Evaluate

11 r + 15 = - 2 r^{2}

Swap the sides

- 2 r^{2} = 11 r + 15

Move the expression to the left side

- 2 r^{2} - 11 r - 15 = 0

Factor the expression

More Steps

Evaluate

- 2 r^{2} - 11 r - 15

Factor out - 1 from the expression

- (2 r^{2} + 11 r + 15)

Factor the expression

More Steps

Evaluate

2 r^{2} + 11 r + 15

Rewrite the expression

2 r^{2} + (5 + 6) r + 15

Calculate

2 r^{2} + 5 r + 6 r + 15

Rewrite the expression

r \times 2 r + r \times 5 + 3 \times 2 r + 3 \times 5

Factor out r from the expression

r (2 r + 5) + 3 \times 2 r + 3 \times 5

Factor out 3 from the expression

r (2 r + 5) + 3 (2 r + 5)

Factor out 2 r + 5 from the expression

(r + 3) (2 r + 5)

- (r + 3) (2 r + 5)

- (r + 3) (2 r + 5) = 0

Divide the terms

(r + 3) (2 r + 5) = 0

When the product of factors equals 0,at least one factor is 0

2 r + 5 = 0 r + 3 = 0

Solve the equation for r

More Steps

Evaluate

2 r + 5 = 0

Move the constant to the right-hand side and change its sign

2 r = 0 - 5

Removing 0 doesn't change the value,so remove it from the expression

2 r = - 5

Divide both sides

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

Divide the numbers

r = \frac{- 5}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

r = - \frac{5}{2}

r = - \frac{5}{2} r + 3 = 0

Solve the equation for r

More Steps

Evaluate

r + 3 = 0

Move the constant to the right-hand side and change its sign

r = 0 - 3

Removing 0 doesn't change the value,so remove it from the expression

r = - 3

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

Solution

r_{1} = - 3, r_{2} = - \frac{5}{2}

Alternative Form

r_{1} = - 3, r_{2} = - 2.5

Show Solution

Rewrite the equation

61 x^{2} + 61 y^{2} = 4 x^{4} + 4 y^{4} + 225 + 8 x^{2} y^{2}

Evaluate

11 r + 15 = - 2 r^{2}

Rewrite the expression

11 r + 2 r^{2} = - 15

Use substitution

More Steps

Evaluate

11 r + 2 r^{2}

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

11 r + 2 (x^{2} + y^{2})

Simplify the expression

11 r + 2 x^{2} + 2 y^{2}

11 r + 2 x^{2} + 2 y^{2} = - 15

Simplify the expression

11 r = - 2 x^{2} - 2 y^{2} - 15

Square both sides of the equation

(11 r)^{2} = (- 2 x^{2} - 2 y^{2} - 15)^{2}

Evaluate

121 r^{2} = (- 2 x^{2} - 2 y^{2} - 15)^{2}

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

121 (x^{2} + y^{2}) = (- 2 x^{2} - 2 y^{2} - 15)^{2}

Evaluate the power

121 (x^{2} + y^{2}) = (2 x^{2} + 2 y^{2} + 15)^{2}

Calculate

121 x^{2} + 121 y^{2} = 4 x^{4} + 4 y^{4} + 225 + 8 x^{2} y^{2} + 60 x^{2} + 60 y^{2}

Move the expression to the left side

121 x^{2} + 121 y^{2} - (60 x^{2} + 60 y^{2}) = 4 x^{4} + 4 y^{4} + 225 + 8 x^{2} y^{2}

Calculate

More Steps

Evaluate

121 x^{2} - 60 x^{2}

Collect like terms by calculating the sum or difference of their coefficients

(121 - 60) x^{2}

Subtract the numbers

61 x^{2}

61 x^{2} + 121 y^{2} = 4 x^{4} + 4 y^{4} + 225 + 8 x^{2} y^{2} + 60 y^{2}

Solution

More Steps

Evaluate

121 y^{2} - 60 y^{2}

Collect like terms by calculating the sum or difference of their coefficients

(121 - 60) y^{2}

Subtract the numbers

61 y^{2}

61 x^{2} + 61 y^{2} = 4 x^{4} + 4 y^{4} + 225 + 8 x^{2} y^{2}

Show Solution

Solve the quadratic equation

Rewrite the equation

Graph