Solve for z square root of 144-24z+z^2=z-12

$\sqrt{144 - 24 z + z^{2}} = z - 12$

Step 1

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

${\sqrt{144 - 24 z + z^{2}}}^{2} = {(z - 12)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{144 - 24 z + z^{2}}$ as ${(144 - 24 z + z^{2})}^{\frac{1}{2}}$ .

${({(144 - 24 z + z^{2})}^{\frac{1}{2}})}^{2} = {(z - 12)}^{2}$

Simplify the left side.

Simplify ${({(144 - 24 z + z^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(144 - 24 z + z^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(144 - 24 z + z^{2})}^{\frac{1}{2} \cdot 2} = {(z - 12)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(144 - 24 z + z^{2})}^{\frac{1}{2} \cdot 2} = {(z - 12)}^{2}$

Rewrite the expression.

${(144 - 24 z + z^{2})}^{1} = {(z - 12)}^{2}$

Simplify.

$144 - 24 z + z^{2} = {(z - 12)}^{2}$

Simplify the right side.

Simplify ${(z - 12)}^{2}$ .

Rewrite ${(z - 12)}^{2}$ as $(z - 12) (z - 12)$ .

$144 - 24 z + z^{2} = (z - 12) (z - 12)$

Expand $(z - 12) (z - 12)$ using the FOIL Method.

Apply the distributive property.

$144 - 24 z + z^{2} = z (z - 12) - 12 (z - 12)$

Apply the distributive property.

$144 - 24 z + z^{2} = z \cdot z + z \cdot - 12 - 12 (z - 12)$

Apply the distributive property.

$144 - 24 z + z^{2} = z \cdot z + z \cdot - 12 - 12 z - 12 \cdot - 12$

Simplify and combine like terms.

Simplify each term.

Multiply $z$ by $z$ .

$144 - 24 z + z^{2} = z^{2} + z \cdot - 12 - 12 z - 12 \cdot - 12$

Move $- 12$ to the left of $z$ .

$144 - 24 z + z^{2} = z^{2} - 12 \cdot z - 12 z - 12 \cdot - 12$

Multiply $- 12$ by $- 12$ .

$144 - 24 z + z^{2} = z^{2} - 12 z - 12 z + 144$

Subtract $12 z$ from $- 12 z$ .

$144 - 24 z + z^{2} = z^{2} - 24 z + 144$

Step 3

Solve for $z$ .

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

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

$144 - 24 z + z^{2} - z^{2} = - 24 z + 144$

Add $24 z$ to both sides of the equation.

$144 - 24 z + z^{2} - z^{2} + 24 z = 144$

Combine the opposite terms in $144 - 24 z + z^{2} - z^{2} + 24 z$ .

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

$144 - 24 z + 0 + 24 z = 144$

Add $144 - 24 z$ and $0$ .

$144 - 24 z + 24 z = 144$

Add $- 24 z$ and $24 z$ .

$144 + 0 = 144$

Add $144$ and $0$ .

$144 = 144$

Since $144 = 144$ , the equation will always be true for any value of $z$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve for z square root of 144-24z+z^2=z-12

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