MYTUTOR SUBJECT ANSWERS

902 views

For what values of k does the graph y=x^(2)+2kx+5 not intersect the x-axis

Where the graph intersects the x-axis, x2+2kx+5 must be equal to zero. Thus we can answer the equivalent question: For what k does x2+2kx+5 = 0 not have a solution?

This is now a simpler problem (roots of a quadratic equation). We can apply the common method of considering the discriminant of x2+2kx+5. Using standard quadratic formula notation where in this case a=1, b =2k and c=5 we evaluate the discriminant : b2-4ac= (2k)2-4*1*5 = 4k2 -20.

Now since the discriminant appears in a square root sign in the quadratic equation, if it is negative there can be no real solutions to the equation ( great this is what we want!).

Thus we want discriminant negative: 4k2 -20 <0. Divide both sides of the inequality by 4 so we have k2-5<0.

Now this is where we must take great care, the following reasoning is a common MISTAKE: rearragne the inequality so we have k2 < 5, then squarrot both sides so we have k < sqrt(5) or k < - sqrt(5) . The second inequalit is implied by the first thus the discriminate negtive for all k values les then the sqrt(5). THIS IS INCORRECT.

When dealing with inequalities involving powers such as we are here we must be extremely careful. the mistake in the reasoning above is when we say k < - sqrt(5), this is actually a form of the common mistake of not inverting the inequality when multiplying both sides of an equation by a negative. Instead when dealing with inequalities with powers it is always much wiser to sketch a graph of the situation.

k2 - 5 is the standard quadratic U shape (think y=x2) shifted down by 5. Having sketched this out it is clear that this graph is less then 0 when it is inbetween it's two roots. 

The roots of k-5 are easy to find: k-5 = 0 implies k2 = 5 implies k = sqrt(5) or k = -sqrt(5).

Comparing this with the graph we can now see that the discriminant is negative for - sqrt(5) < k <  sqrt(5). Thus these are the values for which the graph y=x2+2kx+5 does not intersect the x-axis.

Hugh K. A Level Maths tutor, GCSE Maths tutor, 13 plus  Maths tutor, ...

11 months ago

Answered by Hugh, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

262 SUBJECT SPECIALISTS

£20 /hr

Ioannis P.

Degree: Computer Science (Bachelors) - Warwick University

Subjects offered:Maths

Maths

“Maths and Computer Science are both my passion. Having tutored students in the past, they think that my methods seem very intuitive and natural; you will too.”

Katrina M. IB Maths tutor, 13 Plus  Maths tutor, GCSE Maths tutor, A ...
£20 /hr

Katrina M.

Degree: Mathematics Bsc (Bachelors) - Exeter University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
-Personal Statements-

“Hi, I am a patient, experienced and enthusiastic tutor ready to help each student reach their full potential.”

£36 /hr

James G.

Degree: Mathematical Physics (Doctorate) - Nottingham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.STEP.

“Currently a 3rd year PhD student in Mathematical Physics. I'm very passionate about teaching as well as my subject area. Look forward to hearing from you.”

About the author

Hugh K.

Currently unavailable: for new students

Degree: Mmath - G103 - Mathematics (Masters) - Warwick University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
Chemistry

“About Me: I am a mathematics student at Warwick. I pride myself on having a deep and thorough understanding of mathematical concepts from A level to below. I feel I have encountered the most common classic mistakes mathematics students...”

You may also like...

Posts by Hugh

For what values of k does the graph y=x^(2)+2kx+5 not intersect the x-axis

If cos(x)= 1/3 and x is acute, then find tan(x).

If n is an integer prove (n+3)^(2)-n^(2) is never even.

Other A Level Maths questions

How do you find the inverse of a function?

The radius of a circular disc is increasing at a constant rate of 0.003cm/s. Find the rate at which the area is increasing when the radius is 20cm.

What methods are there for integration?

How do you integrate (sinx)^2?

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok