For the graph of a certain quadratic y = ax^2 + bx + c, the vertex of the parabola is (3,7) and one of the x-intercepts is (-2,0). What is the x-coordinate of the other x-intercept?
Thank you for helping me :D
zeros
x = [-b +/- sqrt(b2-4ac)] /2a
vertex therefore at x = -b/a
3 = -b/2a
x intercepts are the same distance each side of 3
one is five to the left of 3
the other must be 5 to the right
3+5 = 8
Thank you! :)
To find the x-coordinate of the other x-intercept, we first need to determine the equation of the quadratic function based on the given information.
Given that the vertex of the parabola is (3,7), we know that the x-coordinate of the vertex, which is also the axis of symmetry, is x = 3.
Since the x-intercepts occur when y = 0, we can substitute the given x-intercept (-2,0) into the quadratic equation to find the value of a, b, and c.
Substituting x = -2 and y = 0 into the quadratic equation y = ax^2 + bx + c, we get:
0 = a(-2)^2 + b(-2) + c
0 = 4a - 2b + c
Since the vertex of the parabola lies on the axis of symmetry, the x-coordinate of the vertex (-2) is equidistant from both x-intercepts. Therefore, the horizontal distance from the vertex to both x-intercepts is 5 (3 - (-2) = 5). This means the other x-intercept is 5 units to the right of the given x-intercept x = -2.
Using this information, we can set up another equation:
0 = a(5)^2 + b(5) + c
Simplifying, we get:
0 = 25a + 5b + c
Now, we have a system of three equations:
(I) 0 = 4a - 2b + c
(II) 0 = 25a + 5b + c
(III) x = 3 (equation for the axis of symmetry)
To solve this system of equations, we can substitute equation III into equations I and II:
(I) 0 = 4(3) - 2b + c
(II) 0 = 25(3) + 5b + c
Simplifying, we get:
(I) 0 = 12 - 2b + c
(II) 0 = 75 + 5b + c
We can solve this system using substitution or elimination. Subtracting equation (I) from equation (II) eliminates the c term:
0 = 75 + 5b + c - (12 - 2b + c)
0 = 63 + 7b
Simplifying further, we get:
7b = -63
b = -9
Substituting the value of b into equation (I), we find:
0 = 12 - 2(-9) + c
0 = 12 + 18 + c
0 = 30 + c
c = -30
Now that we have the values of a, b, and c, we can write the quadratic equation:
y = ax^2 + bx + c
y = a(x^2 - 9x - 30)
To find the x-coordinate of the other x-intercept, we set y = 0 and solve for x:
0 = a(x^2 - 9x - 30)
Factoring the equation, we get:
0 = (x - 10)(x + 3)
Setting each factor equal to zero, we find the x-intercepts:
x - 10 = 0 ---> x = 10
x + 3 = 0 ---> x = -3
The x-coordinate of the other x-intercept is -3.
Therefore, the x-coordinate of the other x-intercept is -3.