Solve by Factoring square root of 4z-11=z-2

$\sqrt{4 z - 11} = z - 2$

Step 1

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

Subtract $z$ from both sides of the equation.

$\sqrt{4 z - 11} - z = - 2$

Add $2$ to both sides of the equation.

$\sqrt{4 z - 11} - z + 2 = 0$

Step 2

Move all terms not containing $\sqrt{4 z - 11}$ to the right side of the equation.

Add $z$ to both sides of the equation.

$\sqrt{4 z - 11} + 2 = z$

Subtract $2$ from both sides of the equation.

$\sqrt{4 z - 11} = z - 2$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 z - 11}}^{2} = {(z - 2)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 z - 11}$ as ${(4 z - 11)}^{\frac{1}{2}}$ .

${({(4 z - 11)}^{\frac{1}{2}})}^{2} = {(z - 2)}^{2}$

Simplify the left side.

Simplify ${({(4 z - 11)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 z - 11)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 z - 11)}^{\frac{1}{2} \cdot 2} = {(z - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 z - 11)}^{\frac{1}{2} \cdot 2} = {(z - 2)}^{2}$

Rewrite the expression.

${(4 z - 11)}^{1} = {(z - 2)}^{2}$

Simplify.

$4 z - 11 = {(z - 2)}^{2}$

Simplify the right side.

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

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

$4 z - 11 = (z - 2) (z - 2)$

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

Apply the distributive property.

$4 z - 11 = z (z - 2) - 2 (z - 2)$

Apply the distributive property.

$4 z - 11 = z \cdot z + z \cdot - 2 - 2 (z - 2)$

Apply the distributive property.

$4 z - 11 = z \cdot z + z \cdot - 2 - 2 z - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $z$ by $z$ .

$4 z - 11 = z^{2} + z \cdot - 2 - 2 z - 2 \cdot - 2$

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

$4 z - 11 = z^{2} - 2 \cdot z - 2 z - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$4 z - 11 = z^{2} - 2 z - 2 z + 4$

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

$4 z - 11 = z^{2} - 4 z + 4$

Step 5

Solve for $z$ .

Since $z$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$z^{2} - 4 z + 4 = 4 z - 11$

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

Subtract $4 z$ from both sides of the equation.

$z^{2} - 4 z + 4 - 4 z = - 11$

Subtract $4 z$ from $- 4 z$ .

$z^{2} - 8 z + 4 = - 11$

Add $11$ to both sides of the equation.

$z^{2} - 8 z + 4 + 11 = 0$

Add $4$ and $11$ .

$z^{2} - 8 z + 15 = 0$

Factor $z^{2} - 8 z + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$(z - 5) (z - 3) = 0$

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

$z - 5 = 0$

$z - 3 = 0$

Set $z - 5$ equal to $0$ and solve for $z$ .

Set $z - 5$ equal to $0$ .

$z - 5 = 0$

Add $5$ to both sides of the equation.

$z = 5$

Set $z - 3$ equal to $0$ and solve for $z$ .

Set $z - 3$ equal to $0$ .

$z - 3 = 0$

Add $3$ to both sides of the equation.

$z = 3$

The final solution is all the values that make $(z - 5) (z - 3) = 0$ true.

$z = 5, 3$

Solve by Factoring square root of 4z-11=z-2

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