SOLUTION: Find the values of a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 4-√ 17 and 4+√ 17. (Hint: Use the zero-product property in reverse.)

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Question 1192071: Find the values of a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 4-√ 17 and 4+√ 17.
(Hint: Use the zero-product property in reverse.) thank you!
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

and are the two roots of the form and

Where,

Use Vieta's formulas to sum the roots:

and multiply the roots as well

Difference of squares rule

The negative of the sum of the roots is the value of b, while c represents the product of the roots.
b = -8 and c = -1

Why does this work? Consider the two roots to be r and s
This means and lead to and respectively.
Furthermore,
The -(s+r)x term matches with bx, showing that b = -(s+r). We add the two roots, then flip the sign to get the value of b.
The rs term is the product of the roots and matches with c.

Answer:
You can confirm this answer by using the quadratic formula to arrive back to the original roots mentioned.

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Another approach:

Through very nearly identical steps, you should find that also leads to
Using the square root method, solving leads back to the original roots mentioned.

Now let's expand things out and get everything to one side.

We get the same answer as before.

SOLUTION: Find the values of​ a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 4-√ 17 and 4+√ 17. ​(Hint: Use the​ zero-product property in​ reverse.)

SOLUTION: Find the values of a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 4-√ 17 and 4+√ 17. (Hint: Use the zero-product property in reverse.)