Solve for z z = square root of 7z-10

$z = \sqrt{7 z - 10}$

Step 1

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

$\sqrt{7 z - 10} = z$

Step 2

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

${\sqrt{7 z - 10}}^{2} = z^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 z - 10}$ as ${(7 z - 10)}^{\frac{1}{2}}$ .

${({(7 z - 10)}^{\frac{1}{2}})}^{2} = z^{2}$

Simplify the left side.

Simplify ${({(7 z - 10)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(7 z - 10)}^{\frac{1}{2}})}^{2}$ .

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

${(7 z - 10)}^{\frac{1}{2} \cdot 2} = z^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(7 z - 10)}^{\frac{1}{2} \cdot 2} = z^{2}$

Rewrite the expression.

${(7 z - 10)}^{1} = z^{2}$

Simplify.

$7 z - 10 = z^{2}$

Step 4

Solve for $z$ .

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

$7 z - 10 - z^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $7 z - 10 - z^{2}$ .

Reorder the expression.

Move $- 10$ .

$7 z - z^{2} - 10 = 0$

Reorder $7 z$ and $- z^{2}$ .

$- z^{2} + 7 z - 10 = 0$

Factor $- 1$ out of $- z^{2}$ .

$- (z^{2}) + 7 z - 10 = 0$

Factor $- 1$ out of $7 z$ .

$- (z^{2}) - (- 7 z) - 10 = 0$

Rewrite $- 10$ as $- 1 (10)$ .

$- (z^{2}) - (- 7 z) - 1 \cdot 10 = 0$

Factor $- 1$ out of $- (z^{2}) - (- 7 z)$ .

$- (z^{2} - 7 z) - 1 \cdot 10 = 0$

Factor $- 1$ out of $- (z^{2} - 7 z) - 1 (10)$ .

$- (z^{2} - 7 z + 10) = 0$

Factor.

Factor $z^{2} - 7 z + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$- ((z - 5) (z - 2)) = 0$

Remove unnecessary parentheses.

$- (z - 5) (z - 2) = 0$

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

$z - 5 = 0$

$z - 2 = 0$

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

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

$z - 2 = 0$

Add $2$ to both sides of the equation.

$z = 2$

The final solution is all the values that make $- (z - 5) (z - 2) = 0$ true.

$z = 5, 2$

Solve for z z = square root of 7z-10

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