Find the Points of Intersection x^2+y=6 x+y=4

$x^{2} + y = 6$ $x + y = 4$

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

$y = 6 - x^{2}$

$x + y = 4$

Replace all occurrences of $y$ with $6 - x^{2}$ in each equation.

Replace all occurrences of $y$ in $x + y = 4$ with $6 - x^{2}$ .

$x + 6 - x^{2} = 4$

$y = 6 - x^{2}$

Simplify the left side.

Remove parentheses.

$x + 6 - x^{2} = 4$

$y = 6 - x^{2}$

$x + 6 - x^{2} = 4$

$y = 6 - x^{2}$

$x + 6 - x^{2} = 4$

$y = 6 - x^{2}$

Solve for $x$ in $x + 6 - x^{2} = 4$ .

Subtract $4$ from both sides of the equation.

$x + 6 - x^{2} - 4 = 0$

$y = 6 - x^{2}$

Subtract $4$ from $6$ .

$x - x^{2} + 2 = 0$

$y = 6 - x^{2}$

Factor the left side of the equation.

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

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

$- x^{2} + x + 2 = 0$

$y = 6 - x^{2}$

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

$- (x^{2}) + x + 2 = 0$

$y = 6 - x^{2}$

Factor $- 1$ out of $x$ .

$- (x^{2}) - 1 (- x) + 2 = 0$

$y = 6 - x^{2}$

Rewrite $2$ as $- 1 (- 2)$ .

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

$y = 6 - x^{2}$

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

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

$y = 6 - x^{2}$

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

$- (x^{2} - x - 2) = 0$

$y = 6 - x^{2}$

$- (x^{2} - x - 2) = 0$

$y = 6 - x^{2}$

Factor.

Factor $x^{2} - x - 2$ using the AC method.

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

$y = 6 - x^{2}$

Write the factored form using these integers.

$- ((x - 2) (x + 1)) = 0$

$y = 6 - x^{2}$

$- ((x - 2) (x + 1)) = 0$

$y = 6 - x^{2}$

Remove unnecessary parentheses.

$- (x - 2) (x + 1) = 0$

$y = 6 - x^{2}$

$- (x - 2) (x + 1) = 0$

$y = 6 - x^{2}$

$- (x - 2) (x + 1) = 0$

$y = 6 - x^{2}$

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

$x - 2 = 0$

$x + 1 = 0$

$y = 6 - x^{2}$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

$y = 6 - x^{2}$

Add $2$ to both sides of the equation.

$x = 2$

$y = 6 - x^{2}$

$x = 2$

$y = 6 - x^{2}$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

$y = 6 - x^{2}$

Subtract $1$ from both sides of the equation.

$x = - 1$

$y = 6 - x^{2}$

$x = - 1$

$y = 6 - x^{2}$

The final solution is all the values that make $- (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

$y = 6 - x^{2}$

$x = 2, - 1$

$y = 6 - x^{2}$

Replace all occurrences of $x$ with $2$ in each equation.

Replace all occurrences of $x$ in $y = 6 - x^{2}$ with $2$ .

$y = 6 - {(2)}^{2}$

$x = 2$

Simplify the right side.

Simplify $6 - {(2)}^{2}$ .

Simplify each term.

Raise $2$ to the power of $2$ .

$y = 6 - 1 \cdot 4$

$x = 2$

Multiply $- 1$ by $4$ .

$y = 6 - 4$

$x = 2$

$y = 6 - 4$

$x = 2$

Subtract $4$ from $6$ .

$y = 2$

$x = 2$

$y = 2$

$x = 2$

$y = 2$

$x = 2$

$y = 2$

$x = 2$

Replace all occurrences of $x$ with $- 1$ in each equation.

Replace all occurrences of $x$ in $y = 6 - x^{2}$ with $- 1$ .

$y = 6 - {(- 1)}^{2}$

$x = - 1$

Simplify the right side.

Simplify $6 - {(- 1)}^{2}$ .

Simplify each term.

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

Raise $- 1$ to the power of $1$ .

$y = 6 + (- 1) {(- 1)}^{2}$

$x = - 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 6 + {(- 1)}^{1 + 2}$

$x = - 1$

$y = 6 + {(- 1)}^{1 + 2}$

$x = - 1$

Add $1$ and $2$ .

$y = 6 + {(- 1)}^{3}$

$x = - 1$

$y = 6 + {(- 1)}^{3}$

$x = - 1$

Raise $- 1$ to the power of $3$ .

$y = 6 - 1$

$x = - 1$

$y = 6 - 1$

$x = - 1$

Subtract $1$ from $6$ .

$y = 5$

$x = - 1$

$y = 5$

$x = - 1$

$y = 5$

$x = - 1$

$y = 5$

$x = - 1$

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, 2)$

$(- 1, 5)$

The result can be shown in multiple forms.

Point Form:

$(2, 2), (- 1, 5)$

Equation Form:

$x = 2, y = 2$

$x = - 1, y = 5$

Find the Points of Intersection x^2+y=6 x+y=4

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