Full solution

Q. Solve the system of equations.

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y = 21x + 44

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y = x^2 + 3x + 25

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

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(______,______)

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(______,______)

Set Equations Equal: Set the two equations equal to each other since they both equal $y$ . This gives us $21x + 44 = x^2 + 3x + 25$ .
Rearrange and Simplify: Rearrange the equation to set it to zero by subtracting $21x + 44$ from both sides. This gives us $x^2 + 3x + 25 - 21x - 44 = 0$ , which simplifies to $x^2 - 18x - 19 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation $x^2 - 18x - 19 = 0$ . This equation does not factor nicely, so we will use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = -18$ , and $c = -19$ .
Calculate Discriminant: Calculate the discriminant $\Delta = b^2 - 4ac$ which is $\Delta = (-18)^2 - 4(1)(-19)$ . This gives us $\Delta = 324 + 76$ , which simplifies to $\Delta = 400$ .
Use Quadratic Formula: Since the discriminant is positive, we have two real solutions. Calculate the solutions using the quadratic formula: $x = \frac{18 \pm \sqrt{400}}{2}$ . This simplifies to $x = \frac{18 \pm 20}{2}$ .
Find Values of x: Find the two values of x. The first value is $x = (18 + 20) / 2$ which simplifies to $x = 38 / 2$ , giving us $x = 19$ . The second value is $x = (18 - 20) / 2$ which simplifies to $x = -2 / 2$ , giving us $x = -1$ .
Substitute $x = 19$ : Substitute $x = 19$ into the original equation $y = 21x + 44$ to find the corresponding y-value. This gives us $y = 21(19) + 44$ , which simplifies to $y = 399 + 44$ , giving us $y = 443$ .
Substitute $x = -1$ : Substitute $x = -1$ into the original equation $y = 21x + 44$ to find the corresponding y-value. This gives us $y = 21(-1) + 44$ , which simplifies to $y = -21 + 44$ , giving us $y = 23$ .