
How to use the quadratic formula to find roots of a quadratic equation.
Theory:A quadratic equation looks like this: ax² + bx + c = 0Â (where ‘a’ cannot be zero.) Solving the equation means finding ‘x’ values that make the equation true. These ‘x’ values are called the roots Â 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.Â 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... Â Â Â Â Â Â Â Â Â Â Â 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 theÂ x² term, but in that case it is helpful to think of the x² term as 1x² , so then: a = 1,Â Â b = 4,Â Â and c = 5 Substituting these values into the formula we get: Â Â Â Â Â Â Â Â Â Â Â 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:
Â Calculating the square root: Thus:Â Â ,Â Â Â orÂ Â Â
Method: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 1Â Enter the equation to solve and the quadratic formulaClick Â 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
Â button
until the message: ‘INSERT’ appears. Â Â Â Â Â Â
Maths... Â
Â x[2]
– 4x – 5 Â Â Â You will enter the quadratic formula twice, once with a plus sign and once with a minus sign.
Click the Â button, then type (or paste) the quadratic formula with a plus sign: Â Â Â Â Â Â Â Â Â Â Â
Maths... Â
Â x
= (b + (b[2] – 4ac)[1:2]) / (2a) Â
Â Click the Â button, then type (or paste) the quadratic formula with a minus sign: Â Â Â Â Â Â Â Â Â Â Â
Maths... Â
Â x
= (b  (b[2] – 4ac)[1:2]) / (2a) Â
Â Â Click Step 2, Solve...Substitute values for ‘a’, ‘b’, and ‘c’ in these formulas to calculate the roots.
Â For this equation: x² 
4x  5 = 0 we have: a = 1,Â Â Â Â b = 4,Â Â Â andÂ Â Â 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:Â a = 1, b = 4, c = 5
Click Â (substitute) to substitute these values into the formula.
Click Â (simplify) several times to calculate the root. x = 5
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. Â
