Solve for z 9z^2-30z+26=1

$9 z^{2} - 30 z + 26 = 1$

Subtract $1$ from both sides of the equation.

$9 z^{2} - 30 z + 26 - 1 = 0$

Subtract $1$ from $26$ .

$9 z^{2} - 30 z + 25 = 0$

Factor using the perfect square rule.

Rewrite $9 z^{2}$ as ${(3 z)}^{2}$ .

${(3 z)}^{2} - 30 z + 25 = 0$

Rewrite $25$ as $5^{2}$ .

${(3 z)}^{2} - 30 z + 5^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$30 z = 2 \cdot (3 z) \cdot 5$

Rewrite the polynomial.

${(3 z)}^{2} - 2 \cdot (3 z) \cdot 5 + 5^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 z$ and $b = 5$ .

${(3 z - 5)}^{2} = 0$

Set the $3 z - 5$ equal to $0$ .

$3 z - 5 = 0$

Solve for $z$ .

Add $5$ to both sides of the equation.

$3 z = 5$

Divide each term in $3 z = 5$ by $3$ and simplify.

Divide each term in $3 z = 5$ by $3$ .

$\frac{3 z}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{5}{3}$

Divide $z$ by $1$ .

$z = \frac{5}{3}$

The result can be shown in multiple forms.

Exact Form:

$z = \frac{5}{3}$

Decimal Form:

$z = 1.\overline{6}$

Mixed Number Form:

$z = 1 \frac{2}{3}$