Quadratic formula
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How to use the quadratic formula to find roots of a quadratic equation.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to find roots of a quadratic equation by using the quadratic formula.   
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Algematics software. Click here for instructions.


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. 

The quadratic formula is derived from the general quadratic equation (below) by completing the square.

The general quadratic equation...

ax² + bx + c = 0

has roots...


This formula, known as the ‘quadratic formula’, is actually two formulas. The ‘±’ symbol should be read as ‘plus or minus’, which means that you have to work out the formula twice, once with a plus sign in that position, then again with a minus sign.

The first step is to identify the coefficients ‘a’, ‘b’ and ‘c’ in your quadratic equation, so that you can substitute them into the formula to calculate ‘x’.

For this equation:

x² - 4x - 5 = 0

There is no number written in front of the  x² term, but in that case it is helpful to think of the x² term as 1x² , so then:

a = 1,   b = -4,   and c = -5

Substituting these values into the formula we get:


NOTE: If the expression under the square root sign is negative, then there are no real roots and you cannot go any further. You can investigate this before you start by calculating: b² - 4ac

Simplifying the square root term:


 Calculating the square root:

Thus:   ,    or   

Download the free support file... We have created an Algematics document containing the completed example from this topic. It also includes practice exercises to improve your skills.

File name:  'Quadratic formula.alg'   File size: 8kb
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to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.


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 equation to solve and the quadratic formula

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.


   x[2] – 4x – 5



 You will enter the quadratic formula twice, once with a plus sign and once with a minus sign.

NOTE: Instead of typing the formulas below, you can use the Windows clipboard to copy them straight from this web page! First, drag the mouse pointer to select a formula, then hold down a 'Ctrl' key and press the 'C' key. Now in Algematics with the  data entry dialog box displayed, hold down a 'Ctrl' key and press the 'V' key to paste.

Click the  button, then type (or paste) the quadratic formula with a plus sign:


   x = (-b + (b[2] – 4ac)[1:2]) / (2a)



Click the  button, then type (or paste) the quadratic formula with a minus sign:


   x = (-b - (b[2] – 4ac)[1:2]) / (2a)




Step 2, Solve...

Substitute values for ‘a’, ‘b’, and ‘c’ in these formulas to calculate the roots.

These values are found from the quadratic equation as described in the ‘theory’ section at the top of this article, and you need to identify ‘a’, ‘b’, and ‘c’ and write their values down.

If you write a quadratic equation in this form:

ax² + bx + c = 0

then ‘a’ is the number before the x², ‘b’ is the number before the ‘x’, and ‘c’ is the number without an ‘x’. If the x² or ‘x’ terms have no number in front of them, use 1. ‘b’ or ‘c’ will be negative if there is a minus sign before them. If there is no ‘bx’ term, then b = 0, and if there is no ‘c’, then c = 0.


For this equation:

x² - 4x - 5 = 0

we have:

a = 1,     b = -4,    and    c = -5

Click on the first quadratic formula to make it the target.

Click on the input box, and type the values for ‘a’, ‘b’, and ‘c’.

For the example, type:  a = 1, b = -4, c = -5


  a = 1, b = -4, c = -5

Click  (substitute) to substitute these values into the formula.

Click  (simplify) several times to calculate the root.

x = 5
NOTE: The square root term will not simplify unless it is a perfect square. If the square root term is not a perfect square, you can leave it in surd form, or use a pocket calculator to find its approximate value. If the square root term is negative, then there are no real roots and you can't go any further.

Click on the second quadratic formula to make it the target, then follow the same steps as for the first root. For the example quadratic, the other root is x = -1.


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