Answer:
[tex]y=-\frac{5}{7}x^2-\frac{30}{7}x-5[/tex]
Step-by-step explanation:
we know that
The equation of a vertical parabola in factored form is given by
[tex]y=a(x-x_1)(x-x_2)[/tex]
where
a is the leading coefficient
x_1 and x_2 are the x-intercepts
substitute
[tex]y=a(x-(-3-\sqrt{2}))(x-(-3+\sqrt{2}))[/tex]
[tex]y=a(x+(3+\sqrt{2}))(x+(3-\sqrt{2}))[/tex]
Convert to expanded form
[tex]y=a(x^2+x(3-\sqrt{2})+x(3+\sqrt{2})+7)[/tex]
[tex]y=a(x^2+6x+7)[/tex]
Find the value of a
we have the y-intercept (0,-5)
substitute in the equation the value of x and the value of y
[tex]-5=a(0^2+6(0)+7)[/tex]
[tex]-5=7a\\a=-\frac{5}{7}[/tex]
therefore
[tex]y=-\frac{5}{7}(x^2+6x+7)[/tex]
[tex]y=-\frac{5}{7}x^2-\frac{30}{7}x-5[/tex]
By using the given intercepts, we conclude that the equation is:
y = (-5/7)*(x + 3 + √2)*(x + 3 - √2)
We know that a parabola with the x-intercepts (a, 0) and (b, 0) can be written as:
y = a*(x - a)*(x - b).
In this case we know that the x-intercepts are:
(-3 - √2, 0) and (-3 + √2, 0)
Then the equation is:
y = a*(x + 3 + √2)*(x + 3 - √2)
And we also know that the y-intercept is (0, -5), which means that when x = 0, we must have y = -5, replacing that we get:
-5 = a*(3 + √2)*( 3 - √2) = a*(9 - 2) = a*7
-5/7 = a
So we conclude that the equation is:
y = (-5/7)*(x + 3 + √2)*(x + 3 - √2)
If you want to learn more about parabolas, you can read:
https://brainly.com/question/1480401
