Solve for z 5z(z-2)=(z-2)^2

$5 z (z - 2) = {(z - 2)}^{2}$

Step 1

Simplify $5 z (z - 2)$ .

Rewrite.

$0 + 0 + 5 z (z - 2) = {(z - 2)}^{2}$

Simplify by adding zeros.

$5 z (z - 2) = {(z - 2)}^{2}$

Apply the distributive property.

$5 z \cdot z + 5 z \cdot - 2 = {(z - 2)}^{2}$

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$5 (z \cdot z) + 5 z \cdot - 2 = {(z - 2)}^{2}$

Multiply $z$ by $z$ .

$5 z^{2} + 5 z \cdot - 2 = {(z - 2)}^{2}$

Multiply $- 2$ by $5$ .

$5 z^{2} - 10 z = {(z - 2)}^{2}$

Step 2

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

Rewrite ${(z - 2)}^{2}$ as $(z - 2) (z - 2)$ .

$5 z^{2} - 10 z = (z - 2) (z - 2)$

Expand $(z - 2) (z - 2)$ using the FOIL Method.

Apply the distributive property.

$5 z^{2} - 10 z = z (z - 2) - 2 (z - 2)$

Apply the distributive property.

$5 z^{2} - 10 z = z \cdot z + z \cdot - 2 - 2 (z - 2)$

Apply the distributive property.

$5 z^{2} - 10 z = z \cdot z + z \cdot - 2 - 2 z - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $z$ by $z$ .

$5 z^{2} - 10 z = z^{2} + z \cdot - 2 - 2 z - 2 \cdot - 2$

Move $- 2$ to the left of $z$ .

$5 z^{2} - 10 z = z^{2} - 2 \cdot z - 2 z - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$5 z^{2} - 10 z = z^{2} - 2 z - 2 z + 4$

Subtract $2 z$ from $- 2 z$ .

$5 z^{2} - 10 z = z^{2} - 4 z + 4$

Step 3

Move all terms containing $z$ to the left side of the equation.

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

$5 z^{2} - 10 z - z^{2} = - 4 z + 4$

Add $4 z$ to both sides of the equation.

$5 z^{2} - 10 z - z^{2} + 4 z = 4$

Subtract $z^{2}$ from $5 z^{2}$ .

$4 z^{2} - 10 z + 4 z = 4$

Add $- 10 z$ and $4 z$ .

$4 z^{2} - 6 z = 4$

Step 4

Subtract $4$ from both sides of the equation.

$4 z^{2} - 6 z - 4 = 0$

Step 5

Factor the left side of the equation.

Factor $2$ out of $4 z^{2} - 6 z - 4$ .

Factor $2$ out of $4 z^{2}$ .

$2 (2 z^{2}) - 6 z - 4 = 0$

Factor $2$ out of $- 6 z$ .

$2 (2 z^{2}) + 2 (- 3 z) - 4 = 0$

Factor $2$ out of $- 4$ .

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

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

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

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

$2 (2 z^{2} - 3 z - 2) = 0$

Factor.

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 z$ .

$2 (2 z^{2} - 3 z - 2) = 0$

Rewrite $- 3$ as $1$ plus $- 4$

$2 (2 z^{2} + (1 - 4) z - 2) = 0$

Apply the distributive property.

$2 (2 z^{2} + 1 z - 4 z - 2) = 0$

Multiply $z$ by $1$ .

$2 (2 z^{2} + z - 4 z - 2) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((2 z^{2} + z) - 4 z - 2) = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $2 z + 1$ .

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

Remove unnecessary parentheses.

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

Step 6

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

$2 z + 1 = 0$

$z - 2 = 0$

Step 7

Set $2 z + 1$ equal to $0$ and solve for $z$ .

Set $2 z + 1$ equal to $0$ .

$2 z + 1 = 0$

Solve $2 z + 1 = 0$ for $z$ .

Subtract $1$ from both sides of the equation.

$2 z = - 1$

Divide each term in $2 z = - 1$ by $2$ and simplify.

Divide each term in $2 z = - 1$ by $2$ .

$\frac{2 z}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{- 1}{2}$

Divide $z$ by $1$ .

$z = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$z = - \frac{1}{2}$

Step 8

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

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

$z - 2 = 0$

Add $2$ to both sides of the equation.

$z = 2$

Step 9

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

$z = - \frac{1}{2}, 2$

Step 10

The result can be shown in multiple forms.

Exact Form:

$z = - \frac{1}{2}, 2$

Decimal Form:

$z = - 0.5, 2$

Solve for z 5z(z-2)=(z-2)^2

Related Questions