If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value o

If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value of a if b = 5 and c = -6?

If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value of a if b = 5 and c = -6?

Step 1: Identify the roots of the equation, which are given as 3 and -2.
Step 2: Use Vieta's formulas, which tell us that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (3 + -2) equals -b/a and the product of the roots (3 * -2) equals c/a.
Step 3: Calculate the sum of the roots: 3 + (-2) = 1.
Step 4: Calculate the product of the roots: 3 * (-2) = -6.
Step 5: Set up the equations using Vieta's formulas: -b/a = 1 and c/a = -6.
Step 6: Substitute the values of b and c into the equations: -5/a = 1 and -6/a = -6.
Step 7: From -5/a = 1, multiply both sides by a to get -5 = a. Therefore, a = -5.
Step 8: Check the second equation: -6/a = -6. If a = -5, then -6/(-5) = 6/5, which is not equal to -6. This means we need to find a that satisfies both equations.
Step 9: Since the product of the roots gives us c/a = -6, we can rearrange it to find a: a = c / (3 * -2) = -6 / -6 = 1.
Step 10: Therefore, the value of a is 1.

Vieta's Formulas – These formulas relate the coefficients of a polynomial to sums and products of its roots.
Quadratic Equation – Understanding the standard form of a quadratic equation and how to manipulate it to find coefficients.