Question

20 point

Write the equation for the parabola that has x− intercepts (1+
`√(5)` ,0) and (1−
`√(5)` ,0) and passes through the point (4,8).

Answer 1

y=2x^2-4x-8

Explanation:

Answer 2

`y=2x^2-4x-8`

Explanation:

If the parabola has x-intecepts at
`(1+√(5),0)` and
`(1-√(5),0),` then its equation is

`y=a(x-(1+√(5)))(x-(1-√(5)))\\ \\y=a(x-1-√(5))(x-1+√(5))`

This parabola passes through the point (4,8), then its coordinates satisfy the equation:

`8=a(4-1-√(5))(4-1+√(5))\\ \\8=a(3-√(5))(3+√(5))\\ \\8=a(3^2-√(5)^2)\\ \\8=a(9-5)\\ \\8=4a\\ \\a=2`

Hence, parabola's equation is

`y=2(x-1-√(5))(x-1+√(5))\\ \\y=2((x-1)^2-√(5)^2)\\ \\y=2((x-1)^2-5)\\ \\y=2(x^2-2x+1-5)\\ \\y=2(x^2-2x-4)\\ \\y=2x^2-4x-8`