does anyone have a clue how to do this. I am so lost. If you can please explan why you do what you do when working this out so maybe i can better understand this. thanks a bunch.
Prove (algebraically, not numerically) that the product of the solutions of
ax^2+bx+c=0 is c/a
To prove algebraically that the product of the solutions of a quadratic equation ax^2 + bx + c = 0 is c/a, let's break down the steps:
Step 1: Start with the given quadratic equation: ax^2 + bx + c = 0.
Step 2: Solve the quadratic equation using the quadratic formula:
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
Applying this formula, we get:
x = (-b + √(b^2 - 4ac)) / (2a) ---(1)
x = (-b - √(b^2 - 4ac)) / (2a) ---(2)
Step 3: Find the product of the solutions.
Multiply equation (1) and equation (2) together to obtain:
x * x = (-b + √(b^2 - 4ac)) / (2a) * (-b - √(b^2 - 4ac)) / (2a)
Simplifying this expression:
x^2 = (-b + √(b^2 - 4ac)) * (-b - √(b^2 - 4ac)) / (4a^2)
Expanding the expression:
x^2 = (b^2 - (b^2 - 4ac)) / (4a^2)
Simplifying further:
x^2 = (4ac) / (4a^2)
x^2 = c / a^2
Taking the square root of both sides:
x = ± √(c / a^2)
Step 4: Determine the product of the solutions.
The product of the solutions can be found by multiplying the two solutions together:
Product = (√(c / a^2)) * (-√(c / a^2))
Product = (-√(c / a^2)) * (√(c / a^2))
Product = -√(c / a^2) * √(c / a^2)
Product = -c / a
Therefore, the product of the solutions of the quadratic equation ax^2 + bx + c = 0 is -c/a.