Show that the sum of the two solutions of this equaion is -(b/a)

show that the product of the two solutions of this equation is c/a

Results 1 to 8 of 8

- Sep 24th 2007, 06:45 AM #1

- Sep 24th 2007, 07:11 AM #2

- Joined
- Aug 2007
- From
- USA
- Posts
- 3,111
- Thanks
- 2

- Sep 24th 2007, 07:14 AM #3

- Sep 24th 2007, 03:56 PM #4

- Sep 24th 2007, 03:59 PM #5
Let's define $\displaystyle ax^2+bx+c=0,\,\forall a,b,c\in\mathbb R,\,a\ne0$

We all know that the following formula solves the given equation

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

We have two solutions here, so we can set

$\displaystyle x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}\,\wedge\,x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$

Add them, and multiply them, what do you get?

- Sep 24th 2007, 04:02 PM #6

- Sep 24th 2007, 04:40 PM #7

- Joined
- Sep 2007
- Posts
- 127

It's the same as what Plato has done, really - but I'll try to explain it a bit more. (I'm new here, and I don't mean to 'trump' anyone else's answer - so if this is unacceptable, please let me know. Thanks. )

If the roots of $\displaystyle ax^2 + bx + c = 0 $ are $\displaystyle \alpha$, $\displaystyle \beta$

$\displaystyle x = \alpha $ or $\displaystyle \beta$

__Think about what you're doing when you solve a quadratic, and then just do the reverse....__

$\displaystyle (x-\alpha)(x-\beta)=0$

$\displaystyle x^2- \alpha x- \beta x+ \alpha\beta=0$

$\displaystyle x^2-x(\alpha+\beta)+\alpha\beta=0$

(A) $\displaystyle ax^2+bx+c=0$

and (B) $\displaystyle x^2-x(\alpha+\beta)+\alpha\beta=0$

must be the same equation.

__Divide equation (A) by a__, so it becomes $\displaystyle x^2 + \frac{b}{a}x + \frac{c}{a}=0$

__Then compare coefficients:__

x coeff $\displaystyle \rightarrow \alpha+\beta=-\frac{b}{a}$

constant $\displaystyle \rightarrow \alpha\beta=\frac{c}{a}$

- Sep 25th 2007, 04:53 AM #8

- Joined
- Aug 2007
- From
- USA
- Posts
- 3,111
- Thanks
- 2

Original Problem Statements (Emphasis Added)

"Show that the**sum**of the two solutions of this equaion is -(b/a)"

"show that the**product**of the two solutions of this equation is c/a"

"Sum" is addition. "Product" is multiplication.

Original Reply: "A quick look at the Quadratic Formula gives**both**answers.**Write the two solutions separately**and you will see it."

Have you done this, yet?

Write them down. Add them. What is the result?

Write them down again. Myltiply them. What is the result?

Note: $\displaystyle (p+q)(p-q) = p^{2} - q^{2}$