Answer:
[tex]y=2x^2-4x-8[/tex]
Step-by-step explanation:
Factored form of a parabola
[tex]y=a(x-p)(x-q)[/tex]
where:
Given x-intercepts:
Therefore:
[tex]\implies y=a(x-(1+\sqrt{5}))(x-(1-\sqrt{5}))[/tex]
[tex]\implies y=a(x-1-\sqrt{5})(x-1+\sqrt{5})[/tex]
To find a, substitute the given point (4, 8) into the equation and solve for a:
[tex]\implies a(4-1-\sqrt{5})(4-1+\sqrt{5})=8[/tex]
[tex]\implies a(3-\sqrt{5})(3+\sqrt{5})=8[/tex]
[tex]\implies4a=8[/tex]
[tex]\implies a=2[/tex]
Therefore, the equation of the parabola in factored form is:
[tex]\implies y=2(x-1-\sqrt{5})(x-1+\sqrt{5})[/tex]
Expand so that the equation is in standard form:
[tex]\implies y=2(x^2-x+\sqrt{5}x-x+1-\sqrt{5}-\sqrt{5}x+\sqrt{5}-5)[/tex]
[tex]\implies y=2(x^2-x-x+\sqrt{5}x-\sqrt{5}x+\sqrt{5}-\sqrt{5}+1-5)[/tex]
[tex]\implies y=2(x^2-2x-4)[/tex]
[tex]\implies y=2x^2-4x-8[/tex]
