How would I get the two equations into point intercept form?

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

3 years ago

Scott deposits $8000 in an account that earns 6% simple interest. How long will it be before the total amount is $10000?

Ira Lisetskai [31]

A = P(1 + rt)

solve for t

t = ((A/P)-1)/r = ((10000/8000)-1)/0.06 = 4.167 years

8 0

3 years ago

Y varies directly as xy equals 5 when X is 8 how do you find y when X is 9

FrozenT [24]

Given, y varies directly as x

So,

$y=c\cdot x$

Where c is a constant

When y equals 5 when x is 8

So,

$\begin{gathered} 5=c\cdot8 \\ c=\frac{5}{8} \\ \end{gathered}$

So, the relation between y and x

$y=\frac{5}{8}x$

now , when x = 9

Substitute with x to find y

$y=\frac{5}{8}\cdot9=\frac{45}{8}$

So, the answer is option 2. y = 45/8

5 0

1 year ago

Enid's and petter's cars weigh a total of 2.1 tonnes. Edins car weighs 280 kg more than Peter's car

kupik [55]

Answer:

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Step-by-step explanation:

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8 0

3 years ago

You need to fence in a rectangular piece of land for your dog to run and play. The length is 3 times the width. If the perimeter

nikitadnepr [17]

Answer:

Width = 15 feet

Length = 45 feet

Step-by-step explanation:

You need to fence in a rectangular piece of land for your dog to run and play. The length is 3 times the width. If the perimeter is 120 ft, what are the dimensions of your very own dog park?

Perimeter of a rectangle = 2L + 2W

The length is 3 times the width.

L = Length = 3W

W = Width

P = 120 ft

Hence:

120 = 2L + 2W

120 = 2(3W) + 2W

120 = 6W + 2W

120 = 8W

W = 120/8

W = 15 feet

Solving for Length

L = 3W

L = 3 × 15 feet

L = 45 feet

Therefore, the dimensions of your very own dog park is

Width = 15 feet

Length = 45 feet

3 0

3 years ago