hapantad2j

2022-04-22

How do i approach ahead in this question

Let a,$b\in N$ , a is not equal to b and the quadratic equations $(a-1){x}^{2}-({a}^{2}+2)x+{a}^{2}+2a=0\text{}\text{and}\text{}(b-1){x}^{2}-({b}^{2}+2)x+{b}^{2}+2b=0$ have a common root, then the value of ab/5 is

Let a,

Patricia Stanley

Beginner2022-04-23Added 10 answers

Let ${q}_{a}\left(x\right)=(a-1){x}^{2}-({a}^{2}+2)x+{a}^{2}+2a$. Any root r of ${q}_{a}\left(x\right)$ and of ${q}_{b}\left(x\right)$ is also a root of

$(b-1){q}_{a}\left(x\right)-(a-1){q}_{b}\left(x\right)=(-{a}^{2}b+{a}^{2}+a{b}^{2}+2a-{b}^{2}-2b)(x-1).$

Therefore, $r=1$ or

$-{a}^{2}b+{a}^{2}+a{b}^{2}+2a-{b}^{2}-2b=0.tag1$ (1)

But you can't have $r=1$, because

1 is a root of ${q}_{a}\left(x\right)\iff {q}_{a}\left(1\right)=0$

$\iff 3a-3=0$

$\iff a=1$.

So, if 1 was a root of both ${q}_{a}\left(x\right)$ and ${q}_{b}\left(x\right)$, you would have $a=b=1$, but you are assuming that $a\ne b$.

If, on the other hand, you have (1), then $a=b\text{}\text{or}\text{}a=\frac{b+2}{b-1}=1+\frac{3}{b-1}$. Since a, $b\in \mathbb{N}$, this can only occur in two case: when $b=2$ (in which case $a=4$) and when $b=4$ (in which case $a=2$). In both cases, you have $\frac{ab}{5}=\frac{8}{5}$.

Killian Curry

Beginner2022-04-24Added 18 answers

Hint: $t=a$ , b are the two roots of the quadratic $,(t-1){x}^{2}-({t}^{2}+2)x+{t}^{2}+2t=0$ for the value of x which is the common root.

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.