Solve by Factoring 0=x^2-29x+100

$0 = x^{2} - 29 x + 100$

Step 1

Move all the expressions to the left side of the equation.

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

$- x^{2} = - 29 x + 100$

Add $29 x$ to both sides of the equation.

$- x^{2} + 29 x = 100$

Subtract $100$ from both sides of the equation.

$- x^{2} + 29 x - 100 = 0$

Step 2

Factor by grouping.

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 $a \cdot c = - 1 \cdot - 100 = 100$ and whose sum is $b = 29$ .

Factor $29$ out of $29 x$ .

$- x^{2} + 29 (x) - 100 = 0$

Rewrite $29$ as $4$ plus $25$

$- x^{2} + (4 + 25) x - 100 = 0$

Apply the distributive property.

$- x^{2} + 4 x + 25 x - 100 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- x^{2} + 4 x) + 25 x - 100 = 0$

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

$x (- x + 4) - 25 (- x + 4) = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 4$ .

$(- x + 4) (x - 25) = 0$

Step 3

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

$- x + 4 = 0$

$x - 25 = 0$

Step 4

Set $- x + 4$ equal to $0$ and solve for $x$ .

Set $- x + 4$ equal to $0$ .

$- x + 4 = 0$

Solve $- x + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

Step 5

Set $x - 25$ equal to $0$ and solve for $x$ .

Set $x - 25$ equal to $0$ .

$x - 25 = 0$

Add $25$ to both sides of the equation.

$x = 25$

Step 6

The final solution is all the values that make $(- x + 4) (x - 25) = 0$ true.

$x = 4, 25$

Solve by Factoring 0=x^2-29x+100

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