865 views

### How does one show that x^2 + x + 1 > 0 for all values of x?

METHOD 1

Complete the square:

[half the coefficient of the x term, decrease the power of the x^2 and x terms and put in brackets ^2, then correct the final term]

x^2 + x + 1 = (x + 1/2)^2 + 3/4 > 0 for all values of x

since (x + 1/2)^2 > 0 and 3/4 > 0

METHOD 2

Take the derivative of the function f(x) = x^2 + x + 1 and equate to zero:

[for each term, multiply the coefficient by the value of the power and decrease the power by one]

d/dx (x^2 + x + 1) = 2x+1 = 0

Rearrange this to find x:

=> x = -1/2

Plug this into the function:

f(-1/2) = (-1/2)^2 + (-1/2) + 1 = 1/4 – 1/2 + 1 = 3/4

-1/2 < 0 so the function has a minimum point at f(-1/2) (we know that this is a min not max since the function is a quadratic with positive x^2 coefficient). Since this is the only minimum (no. of extrema is one less than the highest power of x, which in this case is 2, so only one extremum) and this minimum is greater than zero, then the function must be greater than zero for all values of x.

f(-1/2) = 3/4 > 0  =>  f(x) > 0 for all values of x

METHOD 3

Determine the discriminant of the function f(x) = x^2 + x + 1

[The discriminant of a function f(x) = ax^2 + bx + c is given by b^2 – 4ac. Here a = 1, b = 1, c = 1]

1^2 – 4*1*1 = -3

The discriminant is less than zero; therefore there are no real routes to the function (i.e. the function does not cross the x-axis). So the function has either a positive minimum or a negative maximum. Since the function is a quadratic with positive x^2 coefficient we know that it must have a positive minimum, therefore the function is positive for all x.

[b^2 – 4ac < 0 => complex conjugate pair of solutions;

b^2 – 4ac = 0 =>  one real (repeated) solution;

b^2 – 4ac > 0 => two real solutions]

1 year ago

Answered by Anya, an A Level Maths tutor with MyTutor

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

#### 423 SUBJECT SPECIALISTS

£20 /hr

Degree: Aeronautics and Astronautics (Integrated Masters) - Southampton University

Subjects offered:Maths, Physics+ 3 more

Maths
Physics
Extended Project Qualification
Chemistry
-Personal Statements-

“Aero and Astro Engineering student specialising in Maths and Physics A-level and GCSE.”

MyTutor guarantee

|  1 completed tutorial

£20 /hr

Degree: Economics and Accounting (Bachelors) - Bristol University

Subjects offered:Maths, Economics

Maths
Economics

“I achieved top grades whilst juggling cricket at a high level. I’ve tutored for Young Einstein Tuition & been a Peer Mentor to those facing personal issues”

£20 /hr

Degree: Chemistry (Bachelors) - University College London University

Subjects offered:Maths, Chemistry

Maths
Chemistry

“Hi, I'm a first year Chemistry student at UCL, I have experience in tutoring and am willing to tutor people of all abilities”

Currently unavailable: for regular students

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths

Maths

“Hi! My name is Anya and I’m a 2nd year Mathematics student at the University of Bristol. I’m afriendly and patient individual, excited for the opportunity to aid pupils in the advancement of their mathematicalfluency and confidence, a...”

### You may also like...

#### Posts by Anya

How does one find the derivative of ln(x)?

How does one find the highest common factor (HCF) and lowest common multiple (LCM) of 36 and 48?

How does one find the inverse of a non-singular 3x3 matrix A?

How does one show that x^2 + x + 1 > 0 for all values of x?

#### Other A Level Maths questions

How to integrate lnx by parts?

Integrate cos^2(x)

Integrate (x)(e^x) with respect to x and then integrate (x)(e^x) with respect to y.

For a given function F(x), what does the graph of the function F(x+2) look like in comparrison to F(x)?

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