If the quadratic equation ax^2 + bx + c = 0 has roots 3 and -2, what is the value of a?

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Q: If the quadratic equation ax^2 + bx + c = 0 has roots 3 and -2, what is the value of a?

Solution: Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.

Steps: 10

Step 1: Understand that the roots of the quadratic equation are the values of x that make the equation equal to zero. In this case, the roots are 3 and -2.
Step 2: Use the formula for the sum of the roots, which is given by -b/a. The sum of the roots (3 + (-2)) is 1.
Step 3: Set up the equation for the sum of the roots: 1 = -b/a.
Step 4: Use the formula for the product of the roots, which is c/a. The product of the roots (3 * -2) is -6.
Step 5: Set up the equation for the product of the roots: -6 = c/a.
Step 6: Choose a value for a. A simple choice is a = 1.
Step 7: Substitute a = 1 into the equations from Steps 3 and 5 to find b and c.
Step 8: From Step 3, if a = 1, then 1 = -b/1, which means b = -1.
Step 9: From Step 5, if a = 1, then -6 = c/1, which means c = -6.
Step 10: Now we have a = 1, b = -1, and c = -6, which satisfies the original quadratic equation.