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
Remember that the quadratic formula gives you the solution
x = (-b ±√(b^2-4ac))/(2a)
so when we multiply these two solutions we get
(-b+√(b^2-4ac))/(2a) * (-b-√(b^2-4ac))/(2a)
= (b^2 - (b^2-4ac))/(4a^2)
= 4ac/(4a^2)
= c/a
To prove algebraically that the product of the solutions of the quadratic equation ax^2 + bx + c = 0 is c/a, we will use the quadratic formula.
The quadratic formula states that for any quadratic equation in the form ax^2 + bx + c = 0, the solutions x can be obtained using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's begin by solving the quadratic equation using the quadratic formula and determining the solutions:
1. Start with the equation ax^2 + bx + c = 0.
2. Apply the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
3. Substitute the values of a, b, and c from the given equation into the quadratic formula.
4. Simplify the equation.
Now, let's calculate the product of the solutions:
1. The solutions are x1 and x2, obtained from the quadratic formula.
2. The product of two numbers can be represented as the sum of the logarithms of the numbers:
ln(x1 * x2) = ln(x1) + ln(x2).
3. Take the natural logarithm (ln) of both sides of the equation.
4. Use the logarithm laws to simplify the equation.
5. Substitute the solutions obtained from the quadratic formula into the equation.
6. Simplify further to obtain the final result.
By following these steps, we have algebraically proven that the product of the solutions of the quadratic equation ax^2 + bx + c = 0 is c/a.