A quadratic equation is graphed. Which of the following linear equations combines with the graphed equation to create a system of equations whose solutions are the points $\left(3, \frac{5}{2}\right)$ and $(-2,-5)$ ?

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Algebra 2

Write a quadratic function from its x-intercepts and another point

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Q. A quadratic equation is graphed. Which of the following linear equations combines with the graphed equation to create a system of equations whose solutions are the points

\left(3, \frac{5}{2}\right)

and

(-2,-5)

Calculate Slope: We need a linear equation in the form $y = mx + b$ that passes through the points $(3, \frac{5}{2})$ and $(-2, -5)$ .
Simplify Slope: First, calculate the slope $m$ using the two points. $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - \left(\frac{5}{2}\right)}{-2 - 3}$
Find Y-Intercept: Simplify the slope calculation. $\newline$ $m = (\frac{-5 - 2.5}{-5}) = \frac{-7.5}{-5} = 1.5$
Solve for $b$ : Now we have the slope, $m = 1.5$ . Next, use one of the points to find $b$ , the y-intercept. $\newline$ Let's use the point $(3, \frac{5}{2})$ . Plug into $y = mx + b$ . $\newline$ $\frac{5}{2} = 1.5(3) + b$
Final Linear Equation: Solve for $b$ . $\frac{5}{2} = 4.5 + b$ $b = \frac{5}{2} - 4.5$
Final Linear Equation: Solve for $b$ . $\frac{5}{2} = 4.5 + b$ $b = \frac{5}{2} - 4.5$ Convert $4.5$ to a fraction to subtract from $\frac{5}{2}$ . $b = \frac{5}{2} - \frac{9}{2}$
Final Linear Equation: Solve for $b$ . $\frac{5}{2} = 4.5 + b$ $b = \frac{5}{2} - 4.5$ Convert $4.5$ to a fraction to subtract from $\frac{5}{2}$ . $b = \frac{5}{2} - \frac{9}{2}$ Solve for $b$ . $b = \frac{5 - 9}{2}$ $b = -\frac{4}{2}$ $b = -2$
Final Linear Equation: Solve for $b$ . $\frac{5}{2} = 4.5 + b$ $b = \frac{5}{2} - 4.5$ Convert $4.5$ to a fraction to subtract from $\frac{5}{2}$ . $b = \frac{5}{2} - \frac{9}{2}$ Solve for $b$ . $b = \frac{5 - 9}{2}$ $b = -\frac{4}{2}$ $b = -2$ We have $\frac{5}{2} = 4.5 + b$ $0$ and $b = -2$ . Write the final linear equation. $\frac{5}{2} = 4.5 + b$ $2$