780 views

### Finding Roots of Quadratic Equations

What is a root to an equation?

A root to an equation is a set of value(s) that satisfy the equation and when shown graphically they are the x values at which the function intercepts the x-axis.

The general form of a quadratic equation is:

ax2 + bx + c = 0                       where a, b and c are real coefficients

and before attempting to solve any quadratic function, you should always aim to get it into this form first.

If it is not in the correct form it can be converted by adding and subtracting each side functions of x in the initial form, for example:

x2 = 2x – 12                              (Subtract both sides by 2x)

x2 – 2x = -12                             (Add 12 to both sides)

x2 – 2x + 12 = 0

As you can now see the previous equation is now in the standard form ax2 + bx + c = 0, where a = 1, b = 2 and c = 12.

Finding the Root(s) of Quadratic Equations

The first way to solve a quadratic equation is by factorising it, an example is:

x2 + 7x + 12 = 0           -->             (x + 3)(x + 4) = 0

the root to the equation is then given by the negative coefficient of the real number inside the bracket, hence is -3 and -4. A sketch of this graph would consist of a U shape intercepting the x-axis at -3 and -4.

The second way to solve a quadratic equation is to complete the square, an example is:

x2 - 10x + 25 = 0           -->             (x-5)2 = 9

The root to this equation is then worked out by square rooting each side and adding 5 to both sides, giving 8 and -2.

The final way to solve them by a quadratic formula, which is:

x = (-b +/- sqr(b2 – 4ac))/2a

The quadratic formula contains the function b2 – 4ac, this is called the discriminant and a, b and c are the coefficients of the equation when in the standard form. The value of the discriminant can show how many roots are present for a particular equation:

b2 – 4ac > 0                              2 real roots

b2 – 4ac = 0                              1 real root

b2 – 4ac < 0                              2 imaginary roots (Complex conjugates)

Example 1

x2 + 6x + 3 = 0                         a=1, b=6 and c=3

b2 – 4ac = 36 – 12 = 24

hence x = (-6 +/- 2sqr6)/2 = -3 +/- sqr6

so the two roots are -3 + sqr6 and -3 – sqr6

Example 2

x2 + 2x + 1 = 0                         a=1, b=2 and c=1

b2 – 4ac = 0

hence  x = -2/2 = -1

Example 3

x2 + 8x + 25 = 0    a=1, b=8 and c=25

b2 – 4ac = 64 – 100 = -36

The discriminant is less than 0, which shows that 2 complex conjugate roots are the solutions to the equation.

Since we can not find the square root of a negative number, we instead denote the term i, this represents the square root of -1 and also shows that i2 = -1. This now allows the solution to be found:

x = (-8 +/- 6i)/2 = -4 +/- 3i

hence the solutions are -4 + 3i and -4 – 3i which are complex conjugates

2 years ago

Answered by Alexander, a GCSE Maths tutor with MyTutor

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

#### 615 SUBJECT SPECIALISTS

£24 /hr

Degree: Mathematical and Theoretical Physics (Masters) - Oxford, Merton College University

Subjects offered:Maths, Science+ 5 more

Maths
Science
Physics
Further Mathematics
Chemistry
.MAT.
-Personal Statements-

“Mathematics and Theoretical Physics, University of Oxford. I enjoy sharing my experience and enthusiasm in Maths with those who could do with some help”

£18 /hr

Degree: Civil and Structural Engineering with a Year in Industry (Masters) - Sheffield University

Subjects offered:Maths, Spanish+ 4 more

Maths
Spanish
Science
Physics
English
Economics

“About Me: Hello! I am a Civil Engineering student at the University of Sheffield. I have always had a natural incline towards all things Maths and Science, so without a doubt, my genuine passion for the subjects’ I am offering will be...”

£18 /hr

Degree: Natural Sciences (Bachelors) - Cambridge University

Subjects offered:Maths, Chemistry+ 2 more

Maths
Chemistry
Biology
-Medical School Preparation-

“I would like to share my passion for science and help you discover the key patterns which govern our world.”

Currently unavailable: until 18/05/2016

Degree: Natural Science specialising in Mathematics, Chemistry and Physics (Masters) - York University

Subjects offered:Maths, Physics

Maths
Physics

“About Me: I'm currently a first year undergraduate at University of York, studying Natural Sciences specializing in Mathematics, Chemistry and Physics. I have a real passion and drive when it comes to all things science and maths.  I...”

MyTutor guarantee

### You may also like...

#### Other GCSE Maths questions

Solve 4x + 6 = 3

What does Pythagoras's theorem state?

Solve the following quadratic equation: x^2 + 3x + 2 = 0

Show that sqrt(20) = 2sqrt(5)

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