Questions: Найди координаты точек пересечения параболы y=x^2+19x-48 и прямой y=9x+96. Запиши в полях ответа найденные решения в порядке возрастания значений по x . ( ; ) ( ; )

Solution Steps

To find the intersection points of the parabola \( y = x^2 + 19x - 48 \) and the line \( y = 9x + 96 \), we set the equations equal to each other:

\[ x^2 + 19x - 48 = 9x + 96 \]

Rearranging the equation gives us:

\[ x^2 + 19x - 9x - 48 - 96 = 0 \]

This simplifies to:

\[ x^2 + 10x - 144 = 0 \]

We solve the quadratic equation \( x^2 + 10x - 144 = 0 \) using the quadratic formula or factoring. The solutions for \( x \) are:

\[ x_1 = -18, \quad x_2 = 8 \]

Next, we find the corresponding \( y \) values for each \( x \) by substituting back into either original equation. Using the parabola equation:

For \( x = -18 \):

\[ y = (-18)^2 + 19(-18) - 48 = 324 - 342 - 48 = -66 \]

For \( x = 8 \):

\[ y = (8)^2 + 19(8) - 48 = 64 + 152 - 48 = 168 \]

The intersection points are:

\[ (-18, -66) \quad \text{and} \quad (8, 168) \]

Finally, we list the intersection points in order of increasing \( x \):

\[ (-18, -66) \quad \text{and} \quad (8, 168) \]

Final Answer