Algebra Examples

$p = - 2 n$ and $p = n^{2} - 3$

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $- 2 n = n^{2} - 3$ for $n$ .

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Step 2.1

Subtract $n^{2}$ from both sides of the equation.

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

Step 2.2

Add $3$ to both sides of the equation.

$- 2 n - n^{2} + 3 = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $- 1$ out of $- 2 n - n^{2} + 3$ .

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Step 2.3.1.1

Reorder $- 2 n$ and $- n^{2}$ .

$- n^{2} - 2 n + 3 = 0$

Step 2.3.1.2

Factor $- 1$ out of $- n^{2}$ .

$- (n^{2}) - 2 n + 3 = 0$

Step 2.3.1.3

Factor $- 1$ out of $- 2 n$ .

$- (n^{2}) - (2 n) + 3 = 0$

Step 2.3.1.4

Rewrite $3$ as $- 1 (- 3)$ .

$- (n^{2}) - (2 n) - 1 \cdot - 3 = 0$

Step 2.3.1.5

Factor $- 1$ out of $- (n^{2}) - (2 n)$ .

$- (n^{2} + 2 n) - 1 \cdot - 3 = 0$

Step 2.3.1.6

Factor $- 1$ out of $- (n^{2} + 2 n) - 1 (- 3)$ .

$- (n^{2} + 2 n - 3) = 0$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $n^{2} + 2 n - 3$ using the AC method.

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Step 2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3 + p = n^{2} - 3$

Step 2.3.2.1.2

Write the factored form using these integers.

$- ((n - 1) (n + 3)) = 0$

Step 2.3.2.2

Remove unnecessary parentheses.

$- (n - 1) (n + 3) = 0$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$n - 1 = 0$

$n + 3 = 0 + p = n^{2} - 3$

Step 2.5

Set $n - 1$ equal to $0$ and solve for $n$ .

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Step 2.5.1

Set $n - 1$ equal to $0$ .

$n - 1 = 0$

Step 2.5.2

Add $1$ to both sides of the equation.

$n = 1$

Step 2.6

Set $n + 3$ equal to $0$ and solve for $n$ .

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Step 2.6.1

Set $n + 3$ equal to $0$ .

$n + 3 = 0$

Step 2.6.2

Subtract $3$ from both sides of the equation.

$n = - 3$

Step 2.7

The final solution is all the values that make $- (n - 1) (n + 3) = 0$ true.

$n = 1, - 3$

Step 3

The solution of the system of equations is all the values that make the system true.

$n = 1 n = - 3$

Step 4

List all of the solutions.

$n = 1$

$n = - 3$