Solve for $z$ . $\newline$ $z^2 - 10z + 21 = 0$ $\newline$ Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas. $\newline$ $z = \_\_$

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Algebra 2

Solve polynomial equations

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Q. Solve for

z

\newline

z^2 - 10z + 21 = 0

\newline

Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.

\newline

z = \_\_

Identify values of $a$ , $b$ , $c$ : Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $z^2 - 10z + 21 = 0$ , which is in the form $az^2 + bz + c = 0$ . Here, $a = 1$ , $b = -10$ , and $b$ $0$ .
Factor the quadratic equation: Factor the quadratic equation. We need to find two numbers that multiply to give $ac$ (which is $21$ ) and add up to give $b$ (which is $-10$ ). $\newline$ The numbers that satisfy these conditions are $-7$ and $-3$ because $(-7) \times (-3) = 21$ and $(-7) + (-3) = -10$ .
Write factored form: Write the factored form of the quadratic equation using the numbers found in Step $2$ . $\newline$ $z^2 - 10z + 21 = (z - 7)(z - 3)$
Solve for z: Set each factor equal to zero and solve for z. $\newline$ First factor: $(z - 7) = 0$ $\newline$ $z = 7$ $\newline$ Second factor: $(z - 3) = 0$ $\newline$ $z = 3$