Solve for n 4n^2+24n+25=4n

$4 n^{2} + 24 n + 25 = 4 n$

Move all terms containing $n$ to the left side of the equation.

Subtract $4 n$ from both sides of the equation.

$4 n^{2} + 24 n + 25 - 4 n = 0$

Subtract $4 n$ from $24 n$ .

$4 n^{2} + 20 n + 25 = 0$

Factor using the perfect square rule.

Rewrite $4 n^{2}$ as ${(2 n)}^{2}$ .

${(2 n)}^{2} + 20 n + 25 = 0$

Rewrite $25$ as $5^{2}$ .

${(2 n)}^{2} + 20 n + 5^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$20 n = 2 \cdot (2 n) \cdot 5$

Rewrite the polynomial.

${(2 n)}^{2} + 2 \cdot (2 n) \cdot 5 + 5^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 n$ and $b = 5$ .

${(2 n + 5)}^{2} = 0$

Set the $2 n + 5$ equal to $0$ .

$2 n + 5 = 0$

Solve for $n$ .

Subtract $5$ from both sides of the equation.

$2 n = - 5$

Divide each term in $2 n = - 5$ by $2$ and simplify.

Divide each term in $2 n = - 5$ by $2$ .

$\frac{2 n}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n}{2} = \frac{- 5}{2}$

Divide $n$ by $1$ .

$n = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{5}{2}$

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{5}{2}$

Decimal Form:

$n = - 2.5$

Mixed Number Form:

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