Solve for x x-x^2+42=0

$x - x^{2} + 42 = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u - u^{2} + 42 = 0$

Factor by grouping.

Reorder terms.

$- u^{2} + u + 42 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 42 = - 42$ and whose sum is $b = 1$ .

Multiply by $1$ .

$- u^{2} + 1 u + 42 = 0$

Rewrite $1$ as $- 6$ plus $7$

$- u^{2} + (- 6 + 7) u + 42 = 0$

Apply the distributive property.

$- u^{2} - 6 u + 7 u + 42 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- u^{2} - 6 u) + 7 u + 42 = 0$

Factor out the greatest common factor (GCF) from each group.

$u (- u - 6) - 7 (- u - 6) = 0$

Factor the polynomial by factoring out the greatest common factor, $- u - 6$ .

$(- u - 6) (u - 7) = 0$

Replace all occurrences of $u$ with $x$ .

$(- x - 6) (x - 7) = 0$

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

$- x - 6 = 0$

$x - 7 = 0$

Set $- x - 6$ equal to $0$ and solve for $x$ .

Set $- x - 6$ equal to $0$ .

$- x - 6 = 0$

Solve $- x - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$- x = 6$

Divide each term in $- x = 6$ by $- 1$ and simplify.

Divide each term in $- x = 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{6}{- 1}$

Divide $x$ by $1$ .

$x = \frac{6}{- 1}$

Simplify the right side.

Divide $6$ by $- 1$ .

$x = - 6$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $(- x - 6) (x - 7) = 0$ true.

$x = - 6, 7$

Solve for x x-x^2+42=0

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