Find the equation of the quadratic given it's real roots $\displaystyle 2-\sqrt 3$ and $\displaystyle 2 + \sqrt3$ which passes through the point (1,-2)

2. if $\displaystyle r_1$ and $\displaystyle r_2$ are two roots of a quadratic ...

$\displaystyle f(x) = k(x - r_1)(x - r_2)$

$\displaystyle f(x) = k[x^2 -(r_1 + r_2)x + r_1 \cdot r_2]$

now, substitute $\displaystyle r_1 = 2-\sqrt{3}$ and $\displaystyle r_2 = 2 + \sqrt{3}$ to determine the basic quadratic in [...], then use the fact that $\displaystyle f(1) = -2$ to find $\displaystyle k$

3. Originally Posted by skeeter
if $\displaystyle r_1$ and $\displaystyle r_2$ are two roots of a quadratic ...

$\displaystyle f(x) = k(x - r_1)(x - r_2)$

$\displaystyle f(x) = k[x^2 -(r_1 + r_2)x + r_1 \cdot r_2]$

now, substitute $\displaystyle r_1 = 2-\sqrt{3}$ and $\displaystyle r_2 = 2 + \sqrt{3}$ to determine the basic quadratic in [...], then use the fact that $\displaystyle f(1) = -2$ to find $\displaystyle k$
I'm not sure exactly what you mean, can you please work it down further? I have never learned this formula.

4. Hello, euclid2!

There are several approaches to this problem.
. . Here's one of them . . .

Find the equation of the quadratic given its real roots $\displaystyle 2-\sqrt 3$ and $\displaystyle 2 + \sqrt3$
which passes through the point (1,-2)

If the quadratic, $\displaystyle ax^2 + bx + c$, has roots $\displaystyle p\text{ and }q$, then: .$\displaystyle p+q \:=\:-\frac{b}{a},\;\;pq \:=\:\frac{c}{a}$

So we have: .$\displaystyle \begin{array}{ccc}p+q &=&4 \\ pq &=& 1\end{array}\quad\Rightarrow\quad\begin{array}{ccc }\frac{b}{a} &=& \text{-}4 \\ \frac{c}{a} &=& 1 \end{array}$ . $\displaystyle \Rightarrow\quad\begin{array}{ccc}b &=&\text{-}4a \\ c &=&a \end{array}$

. . The function (so far) is: .$\displaystyle f(x) \:=\:ax^2 -4ax + a$

Since (1,-2) is on the graph: .$\displaystyle a\!\cdot\!1^2 - 4a\!\cdot\!1 + a \:=\:-2 \quad\Rightarrow\quad -2a \:=\:-2$

. . Hence: .$\displaystyle a\:=\:1,\;\;b\:=\:-4,\;\;c\:=\:1$

Therefore: .$\displaystyle f(x)\;=\;x^2 - 4x + 1$

5. Originally Posted by Soroban
Hello, euclid2!

There are several approaches to this problem.
. . Here's one of them . . .

If the quadratic, $\displaystyle ax^2 + bx + c$, has roots $\displaystyle p\text{ and }q$, then: .$\displaystyle p+q \:=\:-\frac{b}{a},\;\;pq \:=\:\frac{c}{a}$

So we have: .$\displaystyle \begin{array}{ccc}p+q &=&4 \\ pq &=& 1\end{array}\quad\Rightarrow\quad\begin{array}{ccc }\frac{b}{a} &=& \text{-}4 \\ \frac{c}{a} &=& 1 \end{array}$ . $\displaystyle \Rightarrow\quad\begin{array}{ccc}b &=&\text{-}4a \\ c &=&a \end{array}$

. . The function (so far) is: .$\displaystyle f(x) \:=\:ax^2 -4ax + a$

Since (1,-2) is on the graph: .$\displaystyle a\!\cdot\!1^2 - 4a\!\cdot\!1 + a \:=\:-2 \quad\Rightarrow\quad -2a \:=\:-2$

. . Hence: .$\displaystyle a\:=\:1,\;\;b\:=\:-4,\;\;c\:=\:1$

Therefore: .$\displaystyle f(x)\;=\;x^2 - 4x + 1$

Thank you very much, although can you edit this to show me where you are getting those numbers from? Thanks, again.

6. $\displaystyle f(x) = k(x - r_1)(x - r_2)$

$\displaystyle f(x) = k[x^2 -(r_1 + r_2)x + r_1 \cdot r_2]$

$\displaystyle r_1 + r_2 = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 4$

$\displaystyle r_1 \cdot r_2 = (2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1$

$\displaystyle f(x) = k(x^2 - 4x + 1)$

since $\displaystyle f(1) = -2$

$\displaystyle -2 = k[(1)^2 - 4(1) + 1]$

$\displaystyle -2 = k(-2)$

$\displaystyle k = 1$

so ...

$\displaystyle f(x) = x^2 - 4x + 1$