Solve the system of equations. y = x^2 + 41x - 19 y = 42x + 23 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: Set the two equations equal to each other since they both equal y. y = x^2 + 41x - 19 y = 42x + 23 So, x^2 + 41x - 19 = 42x + 23.Subtract and Simplify: Subtract 42x and add 19 to both sides to set the equation to zero. x^2 + 41x - 42x - 19 + 19 = 42x - 42x + 23 + 19 This simplifies to x^2 - x + 42 = 0.Factor Quadratic Equation: Factor the quadratic equation x^2 - x + 42 = 0. This quadratic does not factor nicely, so we will use the quadratic formula to solve for x. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = -1, and c = 42.Calculate Discriminant: Calculate the discriminant (b^2 - 4ac) to determine the nature of the roots. Discriminant = (-1)^2 - 4(1)(42) = 1 - 168 = -167. Since the discriminant is negative, there are no real solutions for x. This means there are no points of intersection.

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Solve the system of equations. $\newline$ $y = x^2 + 41x - 19$ $\newline$ $y = 42x + 23$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Algebra 2

Solve a nonlinear system of equations

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Q. Solve the system of equations.

\newline

y = x^2 + 41x - 19

\newline

y = 42x + 23

\newline

Write the coordinates in exact form. Simplify all fractions and radicals.

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: Set the two equations equal to each other since they both equal $y$ . $y = x^2 + 41x - 19$ $y = 42x + 23$ So, $x^2 + 41x - 19 = 42x + 23$ .
Subtract and Simplify: Subtract $42x$ and add $19$ to both sides to set the equation to zero. $\newline$ $x^2 + 41x - 42x - 19 + 19 = 42x - 42x + 23 + 19$ $\newline$ This simplifies to $x^2 - x + 42 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation $x^2 - x + 42 = 0$ . This quadratic does not factor nicely, so we will use the quadratic formula to solve for $x$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = -1$ , and $c = 42$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots. Discriminant = $(-1)^2 - 4(1)(42) = 1 - 168 = -167$ . Since the discriminant is negative, there are no real solutions for $x$ . This means there are no points of intersection.