For example when solving the inequality:
2x²+x-6>0
we get
(2x-3)(x+2)>0

From this you may be tempted to write x<3/2 and x<-2 however this would be incorrect. The solution must be either -2<3/2, or (x<-2 or x>3/2).
This must be the case for any quadratic with two real roots because all the points between the two roots have one sign and these which are outside the roots have another, as can be clearly seen by considering the general graph.

We can determine which of the two solutions to use by either putting in values to see which work or by drawing a sketch, usually drawing a sketch is the best. For the sketch we only need to consider whether the coefficient of x², in this case 2, is positive or negative so we know which way the graph curves. The y intercept of the graph is irrelevant.

So in our case because 2>=0 the sketch is a “happy” face so the y>0 either side of the roots and therefore our solutions for x are:
x<-2 or x>3/2

By David Woodford

Also see

• Derive Quadratic Formula (16)
• Factorizing Quadratics (1)
• Quadratic Formula (9)

The quadratic formula,

for the general equation ax²+bx+c=0

can be derived using the method of “completing the square” as follows.
starting with

ax²+bx+c=0

divide through by a

x²+bx/a+c/a=0

take the c part to the other side

x²+bx/a=-c/a

In order to complete the square we need to add half the coefficient of x (that’s b/2a) squared to both sides so we get

x²+bx/a + b²/4a²=-c/a +b²/4a²

Now we can factorise the left into a squared bracket (you can check this by multiplying it back out)

(x + b/2a)²=-c/a +b²/4a²

So if we square root both sides and take b/2a to the other side we have x on its own

x = -b/2a ±√(-c/a +b²/4a²)

Now to get this in the more traditional form we can take out 1/2a (Which means each term in the root is timesed by 4a²) from the square root to put the whole thing over 2a

x = (-b ±√(b²-4ac))/2a

Quadratic Solver

Fill in the boxes for the coefficients below to solve any quadratic equation (is also able to find complex roots)

Also see

• Proof of Cosine Rule (3)
• Area of a Triangle (3)
• Proof by Contradiction (0)
• Rational and Irrational Numbers (5)
• How to use Surds – Add, Multiply and Rationalise (1)
• Trigonometry Identities (11)

Note/ sometimes a quadratic cannot be factorized using whole numbers, this is when you must use the quadratic equation to find the values of x. See my earlier post and c++ program

Simple type

Use when there is no coefficent of x²

eg)x²+2x-8

start by opening 2 brackets with an x in each
(x )(x )
put the first sign in the first bracket. If the second sign is + put the same sign in both, if its – put the opposite sign in the second bracket
(x+ )(x- )
find the 2 numbers that will add(if both signs in brackets are +) or subtract(if the signs in the brackets are different) to make the middle number(2) and multiply to make the end number(8)
(x+4)(x-2)
and thats your quadratic factorized

Complex Type

Use when there is a coefficient of x²

eg) 8x²-14x-15
Before we can open brackets we need to split up the 14x
8x² ?x ?x-15
the rule for the signs is the same as in the simple case, put the first sign before the first x term. If the second sign is + put the same sign in both, if its – put the opposite sign before the second x term
8x²-?x+?x-15
we also use the same rule for the to coefficients of x, they must add or subtract to make the middle number(14) but they must times to make the end number times the first(15×8=120)
8x²-20x+6x-15
we then take out the common factor of the first 2 terms(4x)
4x(2x-5) + 6x -15
we use the bracket(2x-5) as the common factor for the second 2 terms and find what we need to multiply by(3)
4x(2x-5)+3(2x-5)
we then take the 2 numbers in front of the brackets(4x and 3) as our second bracket
(4x+3)(2x-5)
and there we have a fully factorized quadratic

Also see

• Derive Quadratic Formula (16)
• Area of a Triangle (3)
• Proof by Contradiction (0)
• Auxiliary Angle Method for Solving Trigonometry Equations (0)
• Rational and Irrational Numbers (5)
• How to use Surds – Add, Multiply and Rationalise (1)

so for the general quadratic
ax²+bx+c=0

So what do you do, well enter a, b and c from the general equation into the formula, work out the answers and they are your solutions.

Whats the plus/minus thingy. You might be wondering what the thing is after the -b, well its a plus or minus sign. Because when you work out a square root it can have to answers, eg)root 9 = +3 and -3, he formula takes this into account by saying you must use the plus and the minus answers. This therefore means you will be 2 solutions to the quadratic, which makes sense as the graph is a curve and it therefore must cut the x-axis twice.

When the root part is negative(before you find the root) there are no real roots, only complex ones. This means that the curve comes down above the x-axis and doesn’t cut it. If you work out the complex roots, using i for root(-1), your pair of answers will be a conjugate pair.

Quadratic Calculator

Enter the coefficients of your quadratic equations into the boxes below and click solve to find the solutions to your quadratic. Will give complex solutions as well if your equation has them.

David Woodfords Quadratic Calculator
This is a console program that i wrote in c++ that will solve a quadratic for you. Just download the file and run it, follow the instructions and it will output the 2 answers for you. I tried to make it work for complex roots as well but somewhere in the decimal data types Ive messed up so only the second parts of the complex answers are correct – though im sure working out -b/2a isn’t too hard for you.
Download quadratic calculator

Also see

• Derive Quadratic Formula (16)
• Auxiliary Angle Method for Solving Trigonometry Equations (0)
• Matrix Calculator, C++ Program by David Woodford (5)
• Factorizing Quadratics (1)
• Simplifying and Solving equations with Algebraic Fractions (0)
• The new Trevor Pythagoras Book Store (0)