The vertex form of a parabola is given by y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Given that the vertex is (-25, 78), we can substitute these values into the equation to get y = a(x + 25)² + 78.

The parabola also intersects the x-axis at two points. This means that the y-coordinate at these points is 0. So, we can set y = 0 in the equation and solve for x to find the x-intercepts.

0 = a(x + 25)² + 78
= a(x + 25)² = -78
= x = ± sqrt(-78/a) - 25

Since the parabola intersects the x-axis at two points, there are two solutions for x. These solutions are the roots of the quadratic equation ax² + bx + c = 0. By comparing this equation with the standard form of a quadratic equation, we can see that the sum of the roots is given by -b/a and the product of the roots is given by c/a.

Therefore, the sum of the constants a, b, and c is given by a + (-b/a) + (c/a) = a - b/a + c/a = a + c - b. Since the roots are symmetric about the vertex, the sum of the roots is 2h = 2*(-25) = -50. Therefore, a + c - b = -50.

So, the possible value of a + b + c is -50.

Question

The vertex form of a parabola is given by y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Given that the vertex is (-25, 78), we can substitute these values into the equation to get y = a(x + 25)² + 78.

The parabola also intersects the x-axis at two points. This means that the y-coordinate at these points is 0. So, we can set y = 0 in the equation and solve for x to find the x-intercepts.

0 = a(x + 25)² + 78
=> a(x + 25)² = -78
=> x = ± sqrt(-78/a) - 25

Since the parabola intersects the x-axis at two points, there are two solutions for x. These solutions are the roots of the quadratic equation ax² + bx + c = 0. By comparing this equation with the standard form of a quadratic equation, we can see that the sum of the roots is given by -b/a and the product of the roots is given by c/a.

Therefore, the sum of the constants a, b, and c is given by a + (-b/a) + (c/a) = a - b/a + c/a = a + c - b. Since the roots are symmetric about the vertex, the sum of the roots is 2h = 2*(-25) = -50. Therefore, a + c - b = -50.

So, the possible value of a + b + c is -50.

Knowee AI · Accepted Answer

The vertex form of a parabola is given by y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Given that the vertex is (-25, 78), we can substitute these values into the equation to get y = a(x + 25)² + 78.

The parabola also intersects the x-axis at two points. This means that the y-coordinate at these points is 0. So, we can set y = 0 in the equation and solve for x to find the x-intercepts.

0 = a(x + 25)² + 78
=> a(x + 25)² = -78
=> x = ± sqrt(-78/a) - 25

Since the parabola intersects the x-axis at two points, there are two solutions for x. These solutions are the roots of the quadratic equation ax² + bx + c = 0. By comparing this equation with the standard form of a quadratic equation, we can see that the sum of the roots is given by -b/a and the product of the roots is given by c/a.

Therefore, the sum of the constants a, b, and c is given by a + (-b/a) + (c/a) = a - b/a + c/a = a + c - b. Since the roots are symmetric about the vertex, the sum of the roots is 2h = 2*(-25) = -50. Therefore, a + c - b = -50.

So, the possible value of a + b + c is -50.

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