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

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Algebra 2

Solve polynomial equations

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Q. Solve for

g

\newline

g^2 - 10g + 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

g = \_\_

Identify values: Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $g^2 - 10g + 21 = 0$ , which are $a = 1$ , $b = -10$ , and $c = 21$ .
Find numbers: Look for two numbers that multiply to $ac$ (which is $1\times21 = 21$ ) and add up to $b$ (which is $-10$ ). The numbers that satisfy these conditions are $-3$ and $-7$ because $-3\times-7 = 21$ and $-3 + -7 = -10$ .
Write factored form: Write the quadratic equation in its factored form using the numbers found in Step $2$ . Since $g^2 - 10g + 21$ factors into $(g - 3)(g - 7)$ , we can set each factor equal to zero to find the solutions for $g$ .
Set first factor: Set the first factor equal to zero: $(g - 3) = 0$ . Solve for $g$ by adding $3$ to both sides of the equation to get $g = 3$ .
Set second factor: Set the second factor equal to zero: $(g - 7) = 0$ . Solve for $g$ by adding $7$ to both sides of the equation to get $g = 7$ .