Solve (2r^2)-9r=5

Question

Solve the quadratic equation

Solve by factoring

Solve using the quadratic formula

Solve by completing the square

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

Alternative Form

r_{1} = - 0.5, r_{2} = 5

Evaluate

2 r^{2} - 9 r = 5

Move the expression to the left side

2 r^{2} - 9 r - 5 = 0

Factor the expression

More Steps

Evaluate

2 r^{2} - 9 r - 5

Rewrite the expression

2 r^{2} + (1 - 10) r - 5

Calculate

2 r^{2} + r - 10 r - 5

Rewrite the expression

r \times 2 r + r - 5 \times 2 r - 5

Factor out r from the expression

r (2 r + 1) - 5 \times 2 r - 5

Factor out - 5 from the expression

r (2 r + 1) - 5 (2 r + 1)

Factor out 2 r + 1 from the expression

(r - 5) (2 r + 1)

(r - 5) (2 r + 1) = 0

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

r - 5 = 0 2 r + 1 = 0

Solve the equation for r

More Steps

Evaluate

r - 5 = 0

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

r = 0 + 5

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

r = 5

r = 5 2 r + 1 = 0

Solve the equation for r

More Steps

Evaluate

2 r + 1 = 0

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

2 r = 0 - 1

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

2 r = - 1

Divide both sides

\frac{2 r}{2} = \frac{- 1}{2}

Divide the numbers

r = \frac{- 1}{2}

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

r = - \frac{1}{2}

r = 5 r = - \frac{1}{2}

Solution

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

Alternative Form

r_{1} = - 0.5, r_{2} = 5

Show Solution

Rewrite the equation

101 x^{2} + 101 y^{2} = 4 x^{4} + 4 y^{4} + 25 + 8 x^{2} y^{2}

Evaluate

(2 r^{2}) - 9 r = 5

Evaluate

2 r^{2} - 9 r = 5

Use substitution

More Steps

Evaluate

2 r^{2} - 9 r

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

2 (x^{2} + y^{2}) - 9 r

Simplify the expression

2 x^{2} + 2 y^{2} - 9 r

2 x^{2} + 2 y^{2} - 9 r = 5

Simplify the expression

- 9 r = - 2 x^{2} - 2 y^{2} + 5

Square both sides of the equation

(- 9 r)^{2} = (- 2 x^{2} - 2 y^{2} + 5)^{2}

Evaluate

81 r^{2} = (- 2 x^{2} - 2 y^{2} + 5)^{2}

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

81 (x^{2} + y^{2}) = (- 2 x^{2} - 2 y^{2} + 5)^{2}

Calculate

81 x^{2} + 81 y^{2} = 4 x^{4} + 4 y^{4} + 25 + 8 x^{2} y^{2} - 20 x^{2} - 20 y^{2}

Move the expression to the left side

81 x^{2} + 81 y^{2} - (- 20 x^{2} - 20 y^{2}) = 4 x^{4} + 4 y^{4} + 25 + 8 x^{2} y^{2}

Calculate

More Steps

Evaluate

81 x^{2} + 20 x^{2}

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

(81 + 20) x^{2}

Add the numbers

101 x^{2}

101 x^{2} + 81 y^{2} = 4 x^{4} + 4 y^{4} + 25 + 8 x^{2} y^{2} - 20 y^{2}

Solution

More Steps

Evaluate

81 y^{2} + 20 y^{2}

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

(81 + 20) y^{2}

Add the numbers

101 y^{2}

101 x^{2} + 101 y^{2} = 4 x^{4} + 4 y^{4} + 25 + 8 x^{2} y^{2}

Show Solution

Solve the quadratic equation

Rewrite the equation

Graph