A monic quadratic has in front of . Its middle coefficient gives the opposite of the roots’ sum, while its constant gives their product. If you’re also told how far apart the roots are, combine the sum and difference to find them. Use the roots’ product for the constant, not their sum.
Hints
- Hint 1
For a monic quadratic, the roots add to the opposite of the -coefficient. What sum does the coefficient give you?
- Hint 2
You know both the roots’ sum and their difference. Add those two equations: the terms involving cancel, leaving an equation for .
- Hint 3
The constant term of a monic quadratic is the roots multiplied together, not their sum. Once you have both roots, what is their product?
Step-by-step
Use the roots’ sum and difference
Step 1Find the roots’ sum
No Desmos needed. The root-coefficient relationship gives the sum, and the stated gap pins down the roots.
A root is a value of that makes the equation equal . For a quadratic with in front of , expanding gives . So the roots’ sum is the opposite of the -coefficient:
- Step 2
Find the first root
The given difference is . Add it to so the terms cancel:
Divide by :
- Step 3
Find the second root
Put into the roots’ sum:
Subtract to find the other root:
- Step 4
Use the roots’ product
The factor pattern shows that the constant term is the product, . Multiply the roots:
So the value of is , not the root sum . Choice C.