Question

Find the x-intercepts of the parabola with

Answer 1

Answer:

The x-intercept are (-7 , 0) and (5 , 0)

Step-by-step 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)