SOLUTION: Write a quadratic equation in standard form with the roots. a) 3+√2 and 3-√2, passes through point (3,1) b) -1+2√3 and -1-2√3, passes through point (-1,4)

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Write a quadratic equation in standard form with the roots. a) 3+√2 and 3-√2, passes through point (3,1) b) -1+2√3 and -1-2√3, passes through point (-1,4)      Log On


   

Question 1118785: Write a quadratic equation in standard form with the roots.
a) 3+√2 and 3-√2, passes through point (3,1)
b) -1+2√3 and -1-2√3, passes through point (-1,4)
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Here is help for (a) only.

y=a%28x-%283%2Bsqrt%282%29%29%29%28x-%283-sqrt%282%29%29%29
y=a%28x-3-sqrt%282%29%29%28x-3%2Bsqrt%282%29%29
y=a%28%28x-3%29-sqrt%282%29%29%28%28x-3%29%2Bsqrt%282%29%29
y=a%28%28x-3%29%5E2-2%29

Solve the formula for a.
a=y%2F%28%28x-3%29%5E2-2%29

Substitute the given point.
a=1%2F%28%283-3%29%5E2-2%29
a=1%2F%28-2%29

Substitute for the now found value of a in the factored form.
y=-%281%2F2%29%28%28x-3%29%5E2-2%29

From that, continue to simplify to get the 'standard' form you want.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The purely algebraic method used by the other tutor is fine; however, with the roots 3+sqrt(2) and 3-sqrt(2), the algebra gets a bit messy.

There is a much easier path to the answers.

In the quadratic equation

y+=+ax%5E2%2Bbx%2Bc

the sum of the roots is -b/a and the product of the roots is c/a. Then we can write the equation as

y+=+a%28x%5E2%2B%28b%2Fa%29x%2B%28c%2Fa%29%29

In your first example, the sum of the roots is 6 and the product is 9-2=7. So an equation of the parabola is

y+=+a%28x%5E2-6x%2B7%29

Then you can find the value of a by substituting the (x,y) values of the given point on the parabola:

1+=+a%283%5E2-6%2A3%2B7%29+=+-2a
a+=+-1%2F2

And the whole equation is

y+=+%28-1%2F2%29%28x%5E2-6x%2B7%29

Then of course use the same method on your second example, where the sum of the roots is -2 and their product is 1-12 = -11.