Solve 110 = (n/2)*(30-1(n-1))

Question

Solve the equation(The real numbers system)

n \in / R

Alternative Form

No real solution

Evaluate

110 = \frac{n}{2} (30 - 1 \times (n - 1))

Simplify

More Steps

Evaluate

\frac{n}{2} (30 - 1 \times (n - 1))

Any expression multiplied by 1 remains the same

\frac{n}{2} (30 - (n - 1))

Subtract the terms

More Steps

Simplify

30 - (n - 1)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

30 - n + 1

Add the numbers

31 - n

\frac{n}{2} (31 - n)

Multiply the terms

\frac{n ( 31 - n )}{2}

110 = \frac{n ( 31 - n )}{2}

Swap the sides

\frac{n ( 31 - n )}{2} = 110

Rewrite the expression

\frac{31}{2} n - \frac{1}{2} n^{2} = 110

Move the expression to the left side

\frac{31}{2} n - \frac{1}{2} n^{2} - 110 = 0

Rewrite in standard form

- \frac{1}{2} n^{2} + \frac{31}{2} n - 110 = 0

Divide both sides

\frac{31 n - n ^{2}}{- \frac{1}{2}} = \frac{220}{- \frac{1}{2}}

Evaluate

- 62 n + 2 n^{2} = - 440

Add the same value to both sides

- 62 n + 2 n^{2} + \frac{961}{4} = - 440 + \frac{961}{4}

Simplify the expression

(n - \frac{31}{2})^{2} = - \frac{799}{4}

Solution

n \in / R

Alternative Form

No real solution

Show Solution

Solve the equation(The complex numbers system)

n_{1} = \frac{31}{2} - \frac{799}{2} i, n_{2} = \frac{31}{2} + \frac{799}{2} i

Evaluate

110 = \frac{n}{2} (30 - 1 \times (n - 1))

Simplify

More Steps

Evaluate

\frac{n}{2} (30 - 1 \times (n - 1))

Any expression multiplied by 1 remains the same

\frac{n}{2} (30 - (n - 1))

Subtract the terms

More Steps

Simplify

30 - (n - 1)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

30 - n + 1

Add the numbers

31 - n

\frac{n}{2} (31 - n)

Multiply the terms

\frac{n ( 31 - n )}{2}

110 = \frac{n ( 31 - n )}{2}

Swap the sides

\frac{n ( 31 - n )}{2} = 110

Multiply both sides

n (31 - n) = 220

Expand the expression

More Steps

Evaluate

n (31 - n)

Apply the distributive property

n \times 31 - n \times n

Use the commutative property to reorder the terms

31 n - n \times n

Multiply the terms

31 n - n^{2}

31 n - n^{2} = 220

Divide both sides

\frac{31 n - n ^{2}}{- \frac{1}{2}} = \frac{220}{- \frac{1}{2}}

Evaluate

- 62 n + 2 n^{2} = - 440

Add the same value to both sides

- 62 n + 2 n^{2} + \frac{961}{4} = - 440 + \frac{961}{4}

Simplify the expression

(n - \frac{31}{2})^{2} = - \frac{799}{4}

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

n - \frac{31}{2} = \pm - \frac{799}{4}

Simplify the expression

n - \frac{31}{2} = \pm \frac{799}{2} i

Separate the equation into 2 possible cases

n - \frac{31}{2} = \frac{799}{2} i n - \frac{31}{2} = - \frac{799}{2} i

Solve the equation

More Steps

Evaluate

n - \frac{31}{2} = \frac{799}{2} i

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

n = \frac{799}{2} i + \frac{31}{2}

Calculate

n = \frac{31}{2} + \frac{799}{2} i

n = \frac{31}{2} + \frac{799}{2} i n - \frac{31}{2} = - \frac{799}{2} i

Solve the equation

More Steps

Evaluate

n - \frac{31}{2} = - \frac{799}{2} i

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

n = - \frac{799}{2} i + \frac{31}{2}

Calculate

n = \frac{31}{2} - \frac{799}{2} i

n = \frac{31}{2} + \frac{799}{2} i n = \frac{31}{2} - \frac{799}{2} i

Solution

n_{1} = \frac{31}{2} - \frac{799}{2} i, n_{2} = \frac{31}{2} + \frac{799}{2} i

Show Solution

Solve the quadratic equation

Solve by factoring

Solve using the quadratic formula

Solve using the PQ formula

n_{1} = 11, n_{2} = 20

Evaluate

110 = \frac{n}{2} (30 - 1 \times (n - 1))

Simplify

More Steps

Evaluate

\frac{n}{2} (30 - 1 \times (n - 1))

Any expression multiplied by 1 remains the same

\frac{n}{2} (30 - (n - 1))

Subtract the terms

More Steps

Simplify

30 - (n - 1)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

30 - n + 1

Add the numbers

31 - n

\frac{n}{2} (31 - n)

Multiply the terms

\frac{n ( 31 - n )}{2}

110 = \frac{n ( 31 - n )}{2}

Swap the sides

\frac{n ( 31 - n )}{2} = 110

Multiply both sides of the equation by LCD

\frac{n ( 31 - n )}{2} \times 2 = 110 \times 2

Simplify the equation

More Steps

Evaluate

\frac{n ( 31 - n )}{2} \times 2

Simplify

n (31 - n)

Apply the distributive property

n \times 31 - n \times n

Use the commutative property to reorder the terms

31 n - n \times n

Multiply the terms

31 n - n^{2}

31 n - n^{2} = 110 \times 2

Simplify the equation

31 n - n^{2} = 220

Move the expression to the left side

31 n - n^{2} - 220 = 0

Factor the expression

More Steps

Evaluate

31 n - n^{2} - 220

Reorder the terms

- n^{2} + 31 n - 220

Rewrite the expression

- n^{2} + (11 + 20) n - 220

Calculate

- n^{2} + 11 n + 20 n - 220

Rewrite the expression

- n \times n + n \times 11 + 20 n - 20 \times 11

Factor out - n from the expression

- n (n - 11) + 20 n - 20 \times 11

Factor out 20 from the expression

- n (n - 11) + 20 (n - 11)

Factor out n - 11 from the expression

(- n + 20) (n - 11)

(- n + 20) (n - 11) = 0

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

- n + 20 = 0 n - 11 = 0

Solve the equation for n

More Steps

Evaluate

- n + 20 = 0

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

- n = 0 - 20

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

- n = - 20

Change the signs on both sides of the equation

n = 20

n = 20 n - 11 = 0

Solve the equation for n

More Steps

Evaluate

n - 11 = 0

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

n = 0 + 11

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

n = 11

n = 20 n = 11

Solution

n_{1} = 11, n_{2} = 20

Show Solution

Solve the equation(The real numbers system)

Solve the equation(The complex numbers system)

Solve the quadratic equation

Graph