Solve for n 4n^2+3=75

$4 n^{2} + 3 = 75$

Step 1

Move all terms not containing $n$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$4 n^{2} = 75 - 3$

Subtract $3$ from $75$ .

$4 n^{2} = 72$

Step 2

Divide each term in $4 n^{2} = 72$ by $4$ and simplify.

Divide each term in $4 n^{2} = 72$ by $4$ .

$\frac{4 n^{2}}{4} = \frac{72}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 n^{2}}{4} = \frac{72}{4}$

Divide $n^{2}$ by $1$ .

$n^{2} = \frac{72}{4}$

Simplify the right side.

Divide $72$ by $4$ .

$n^{2} = 18$

Step 3

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$n = \pm \sqrt{18}$

Step 4

Simplify $\pm \sqrt{18}$ .

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$n = \pm \sqrt{9 (2)}$

Rewrite $9$ as $3^{2}$ .

$n = \pm \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$n = \pm 3 \sqrt{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$n = 3 \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$n = - 3 \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$n = 3 \sqrt{2}, - 3 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$n = 3 \sqrt{2}, - 3 \sqrt{2}$

Decimal Form:

$n = 4.24264068 \dots, - 4.24264068 \dots$

Solve for n 4n^2+3=75

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