Answer:
The x-intercept are (-7 , 0) and (5 , 0)
Explanation:
* Lets revise the general form of the equation of the parabola
- The general form of the equation of the parabola is:
y = ax² + bx + c , a , b , c are constant
- The y-intercept is c, because the parabola intersect the y-axis at
point (0 , c)
- The x-coordinate of the vertex point is -b/2a
* Now lets solve the problem
∵ The general form of the equation of the parabola is y = ax² + bx + c
∵ The y-intercept is -105
∴ c = -105
∴ y = ax² + bx - 105
∵ The vertex point is (-1 , -108)
∴ The x-coordinate of the vertex of the parabola = -1
∵ The x-coordinate of the vertex of the parabola = -b/a
∴ -b/2a = -1 ⇒ using cross multiplication
∴ -b = -2a ⇒ multiply two sides by -1
∴ b = 2a
- Substitute the value of b in the equation
∴ y = ax² + 2ax - 105
- Substitute the value of x , y in the equation by the coordinates of
the vertex point
∵ The vertex point lies on the parabola
∴ put x = -1 and y = -108
∴ -108 = a(-1)² + 2a(-1) - 105
∴ -108 = a - 2a - 105 ⇒ add the like term
∴ -108 = -a - 105 ⇒ add 105 to both sides
∴ -3 = -a ⇒ multiply both sides by -1
∴ a = 3
- Substitute the value of a in the equation
∵ y = ax² + 2ax - 105
∴ y = 3x² + 2(3)x - 105
∴ y = 3x² + 6x - 105
- To find the x-intercept put y = 0
∴ 3x² + 6x - 105 = 0
- All the terms have 3 as a common factor
∴ divide all the terms by 3
∴ x² + 2x - 35 = 0
- Now factorize it into two factors
∵ x² = x × x ⇒ the 1st term in the bracket and the 1st term in the
2nd bracket
∵ -35 = -5 × 7 ⇒ the 2nd term in the 1st bracket and the 2nd term in the
2nd bracket
∵ x × - 5 = -5x ⇒ means
∵ x × 7 = 7x ⇒ extremes
∵ 7x - 5x = 2x ⇒ the middle term
∴ (x - 5)(x + 7) = 0
- Equate each bracket by 0
∴ x - 5 = 0 ⇒ add 5 to both sides
∴ x = 5
OR
∴ x + 7 = 0 ⇒ subtract 7 from both sides
∴ x = -7
∴ The x-intercept are -7 , 5
* The x-intercept are (-7 , 0) and (5 , 0)