Answer:
* The x-intercepts are -7 and 5
Explanation:
* At first lets revise the standard and general forms of the
quadratic function which represented graphically by the parabola
- f(x) = a(x - h)² + k ⇒ standard form
- Where point (h , k) is the vertex of the parabola
- f(x) = ax² + bx + c ⇒ general form
- Where a, b, c are constant
- c is the y-intercept ⇒ means x = 0
- h = -b/2a
- k = f(h)
* Lets solve the problem
- We will find the equation of the parabola
∵ The vertex is (-1 , -108)
∴ h = -1 and k = -108
∵ y-intercept = -105
- Equate the two forms
∵ ax² + bx + c = a(x - h)² + k ⇒ solve the ( )²
∴ ax² + bx + c = a(x² - 2hx + h²) + k ⇒ open the bracket
∴ ax² + bx + c = ax² - 2ahx + ah² + k ⇒ by comparing the two sides
∴ c = ah² + k
- Substitute the value of c , h , k in it
∴ -105 = a(-1)² + -108
∴ -105 = a - 108 ⇒ add 108 to the both sides
∴ 3 = a
- Lets write the equation in the standard form
∴ y = 3(x - -1)² + -108
∴ y = 3(x + 1)² - 108
* To find the x-intercepts means the parabola intersects the x-axis,
then put y = 0
∴ 3(x + 1)² - 108 = 0 ⇒ Add 108 to the both sides
∴ 3(x + 1)² = 108 ⇒ divide the both sides by 3
∴ (x + 1)² = 36 ⇒ take square root for both sides
∴ (x + 1) = ± 6
# x + 1 = 6 OR x + 1 = -6
∵ x + 1 = 6 ⇒ subtract 1 from both sides
∴ x = 5
∵ x + 1 = -6 ⇒ subtract 1 from both sides
∴ x = -7
* The x-intercepts are -7 and 5
∴