Solve n^2-n 1<10620

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for n

\frac{- 42481 + 1}{2} < n < \frac{42481 + 1}{2}

Alternative Form

n \in (\frac{- 42481 + 1}{2}, \frac{42481 + 1}{2})

Evaluate

n^{2} - n \times 1 < 10620

Any expression multiplied by 1 remains the same

n^{2} - n < 10620

Move the expression to the left side

n^{2} - n - 10620 < 0

Rewrite the expression

n^{2} - n - 10620 = 0

Add or subtract both sides

n^{2} - n = 10620

Add the same value to both sides

n^{2} - n + \frac{1}{4} = 10620 + \frac{1}{4}

Simplify the expression

(n - \frac{1}{2})^{2} = \frac{42481}{4}

Take the root of both sides of the equation and remember to use both positive and negative roots

n - \frac{1}{2} = \pm \frac{42481}{4}

Simplify the expression

n - \frac{1}{2} = \pm \frac{42481}{2}

Separate the equation into 2 possible cases

n - \frac{1}{2} = \frac{42481}{2} n - \frac{1}{2} = - \frac{42481}{2}

Solve the equation

More Steps

n = \frac{42481 + 1}{2} n - \frac{1}{2} = - \frac{42481}{2}

Solve the equation

More Steps

n = \frac{42481 + 1}{2} n = \frac{- 42481 + 1}{2}

Determine the test intervals using the critical values

n < \frac{- 42481 + 1}{2} \frac{- 42481 + 1}{2} < n < \frac{42481 + 1}{2} n > \frac{42481 + 1}{2}

Choose a value form each interval

n_{1} = - 104 n_{2} = 1 n_{3} = 105

To determine if n < \frac{- 42481 + 1}{2} is the solution to the inequality,test if the chosen value n = - 104 satisfies the initial inequality

More Steps

n < \frac{- 42481 + 1}{2} is not a solution n_{2} = 1 n_{3} = 105

To determine if \frac{- 42481 + 1}{2} < n < \frac{42481 + 1}{2} is the solution to the inequality,test if the chosen value n = 1 satisfies the initial inequality

More Steps

n < \frac{- 42481 + 1}{2} is not a solution \frac{- 42481 + 1}{2} < n < \frac{42481 + 1}{2} is the solution n_{3} = 105

To determine if n > \frac{42481 + 1}{2} is the solution to the inequality,test if the chosen value n = 105 satisfies the initial inequality

More Steps

n < \frac{- 42481 + 1}{2} is not a solution \frac{- 42481 + 1}{2} < n < \frac{42481 + 1}{2} is the solution n > \frac{42481 + 1}{2} is not a solution

Solution

\frac{- 42481 + 1}{2} < n < \frac{42481 + 1}{2}

Alternative Form

n \in (\frac{- 42481 + 1}{2}, \frac{42481 + 1}{2})

Show Solution