Given that the roots of a quadratic equation $y = x^2 + ax + b$ are -3 and 2, find the value of a and b.
Understand the Problem
The question provides a quadratic equation $y=x^2 + ax + b$ and tells us that its roots are -3 and 2. The task is to find the values of the coefficients 'a' and 'b'. We can approach this either substituting the roots into the quadratic equation, or using Vieta's formulas.
Answer
$a = 1$ $b = -6$
Answer for screen readers
$a = 1$ $b = -6$
Steps to Solve
- Apply Vieta's formulas
Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation of the form $x^2 + ax + b = 0$, where the roots are $r_1$ and $r_2$, the formulas are:
- Sum of the roots: $r_1 + r_2 = -a$
- Product of the roots: $r_1 \cdot r_2 = b$
- Calculate the sum of the roots
The roots are given as -3 and 2. So, their sum is: $r_1 + r_2 = -3 + 2 = -1$
- Find the value of a
Using Vieta's formula for the sum of the roots, we have: $-a = -1$ $a = 1$
- Calculate the product of the roots
The roots are -3 and 2. So, their product is: $r_1 \cdot r_2 = (-3) \cdot (2) = -6$
- Find the value of b
Using Vieta's formula for the product of the roots, we have: $b = -6$
$a = 1$ $b = -6$
More Information
Vieta's formulas offer a shortcut. If the roots were complex or irrational, direct substitution might be more cumbersome than using these formulas.
Tips
A common mistake is getting the sign wrong when applying Vieta's formulas, especially with the sum of the roots. Remember that the sum of the roots is equal to the negative of the coefficient of the $x$ term (which is $a$ in this case) when the coefficient of the $x^2$ term is 1.
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