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Find the set of values of k for which the line y=kx-4 intersects the curve y=x²-2x at 2 distinct points?

Sonbull [250]

Answer:

$-6 < k < 2$

Step-by-step explanation:

Given

$y = x^2 - 2x$

$y =kx -4$

Required

Possible values of k

The general quadratic equation is:

$ax^2 + bx + c = 0$

Subtract $y = x^2 - 2x$ and $y =kx -4$

$y - y = x^2 - 2x - kx +4$

$0 = x^2 - 2x - kx +4$

Factorize:

$0 = x^2 +x(-2 - k) +4$

Rewrite as:

$x^2 +x(-2 - k) +4=0$

Compare the above equation to: $ax^2 + bx + c = 0$

$a = 1$

$b= -2-k$

$c =4$

For the equation to have two distinct solution, the following must be true:

$b^2 - 4ac > 0$

So, we have:

$(-2-k)^2 -4*1*4>0$

$(-2-k)^2 -16>0$

Expand

$4 +4k+k^2-16>0$

Rewrite as:

$k^2 + 4k - 16 + 4 >0$

$k^2 + 4k - 12 >0$

Expand

$k^2 + 6k-2k - 12 >0$

Factorize

$k(k + 6)-2(k + 6) >0$

Factor out k + 6

$(k -2)(k + 6) >0$

Split:

$k -2 > 0$ or $k + 6> 0$

So:

$k > 2$ or k $> -6$

To make the above inequality true, we set:

$k < 2$ or $k >-6$

So, the set of values of k is:

$-6 < k < 2$

3 0

2 years ago

The sum of the digits of a two-digit number is 8. If the digits are interchanged ,new number is 10 more than double the original

vova2212 [387]

Answer:

Original number 26.

Step-by-step explanation:

xy - two-digit number

1) x + y = 8

2) Original two-digit number can be written as

10*x + y

3) If the digits interchanged yx,

then the new number can be written as

10*y + x

4) Double the original number is

2*(10*x + y)

5) New number is 10 more than double the original number

(10*y + x) - (2*(10*x + y)) = 10

6) Now we have the system of 2 equations:

x + y = 8

(10*y + x) - (2*(10*x + y)) = 10 -----> 10y + x - (20x + 2y) = 10 ---> 8y - 19x = 10

x = 8 - y

8y - 19(8 - y) = 10

8y - 152 +19y = 10

27y = 162

y = 6

x = 8 - y = 8 - 6 = 2

x = 2

So, x =2, y = 6.

Original number 26.

Check:

Original number 26.

New number 62.

Double of the original number = 2*26= 52.

New number is 10 more than double the original number :

62 - 52 = 10 True

6 0

3 years ago

Please help me on this geometry i’ll mark u the brainliest

kicyunya [14]

Answer:

Step-by-step explanation:

If XY is bisector of angle AXN then: $\dfrac{YN}{AY}=\dfrac{XN}{AX}$

$\dfrac4{12}=\dfrac{XN}{18}\\\\XN=\dfrac4{12}\cdot18\\\\XN=6$

5 0

3 years ago

A five inch tall bamboo shoot doubles in height every three days. If the equation y=ab^x, where x is the number of doubling peri

stiv31 [10]

The correct answer for the question that is being presented above is this one: "x = 2; a = 5; b = 3" A 5 inch tall bamboo shoot doubles in height every 3 days. If the equation y=ab^x, where x is the number of doubling periods, represents the height of the bamboo shoot

4 0

3 years ago

Você precisa colocar suas renas, Raposa, Trovão, Cometa e Relâmpago, em uma fila única para puxar o seu trenó. No entanto, Comet

Gennadij [26K]

Answer:

If the reindeer do not fight, there are 5!=120 ways to line them up in a straight line.

Out of these, Bloopin and Rudy are together in 4!=24 times. Similarly Rudy and Bloopin will be together 4!=24 times. So the number of ways they are not together is 120-24-24=72 ways.

Hope this helps. :)

Step-by-step explanation:

Spanish:

Si los renos no luchan, hay 5! = 120 formas de alinearlos en línea recta.

De estos, Bloopin y Rudy están juntos en 4! = 24 veces. De manera similar, Rudy y Bloopin estarán juntos 4! = 24 veces. Entonces, el número de formas en que no están juntas es 120-24-24 = 72 formas.

Espero que esto ayude :)

4 0

2 years ago