How to find roots of a quadratic equation by factorising.



Part 1

A quadratic equation looks like this:

ax² + bx + c = 0 (where ‘a’ cannot be zero.)

Solving the equation means finding ‘x’ values that make the equation true. These ‘x’ values are called the roots of the quadratic.

Quadratic equations can have 0, 1 or two roots.

NOTE: In the complex number system, all quadratic equations have roots, but we will not discuss complex numbers in this article. Roots of quadratics always come in pairs, but when there are two roots that are the same we say that there is only one root.

This method requires that you can factorise the quadratic expression on the left hand side. This is not always possible, and if not you would have to use one of the other methods.

Consider this quadratic expression…

x² – 4x – 5 = 0

When factorised, it looks like this…

(x + 1)(x – 5) = 0

When the quadratic expression is factorised, it is written as the product of two factors,
like this:

pq = 0

This equation is true only if either “p” or “q” is zero.
In the example, “p” is (x+1), and “q” is (x-5).

If “p” is zero, then we have: 0*q = 0 which is true,

and if “q” is zero, then we have: p*0 = 0 which is also true.

For the example, this means that if (x+1) or (x-5) is zero, the product will be zero and the equation will be true. We use this fact to find the roots as follows:

x + 1 = 0 so x = -1


x – 5 = 0 so x = 5

The example quadratic equation has two roots, x = -1 and x = 5.


Part 2

IMPORTANT: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the ‘Getting Started’ section of the Algematics program ‘Help’.

Step 1 Enter the data

Click and type the quadratic equation into the ‘maths’ box in the data entry dialog box.

If the ‘EMPTY’ message is not displayed between the blue buttons, click the button until the message: ‘INSERT’ appears.

If necessary, use the ‘ * ‘ symbol for multiply and the ‘ / ‘ symbol for divide.


x[2] – 4x – 5


Step 2 Solve…

Keep clicking the (factorise) button until the equation is factorised, and looks like this:

(x + 1)(x – 5) = 0

If there is a common factor, this will be at the front of the left bracket.

The expressions in brackets must be equated to zero to find the roots. (See the theory section at the top of this article.)

In this case, x+1 = 0 and x-5 = 0, so the roots are x = -1 and x = 5.

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