Solving quadratic equations
Posted in Maths, Core 1A quadratic equation is one in which the highest power of x is two. To solve a quadratic equation, f(x) you are looking for a value, a, such that f(a) = 0. This is often referred to as the 'root' of the equation - it is where the line of the equation on the graph crosses the x axis. There 3 methods to solve quadratic equations.
Completing the square
You may be asked to solve a quadratic equation by 'completing the square' or asked to put a quadratic in the form of (x + a)2 + b, this is pretty simple:
Question: Solve, by completing the square, x2 + 8x + 10
Worked answer: Firstly you want to write the equation in the form of (x + a)2 + b. To find a just divide the coefficient of x by 2, in this case that is 4. This gives the first half of the equation:
(x + 4)2
Expanding this out gives: x2 + 8x + 16. Comparing this with the original equation in the question shows we need to remove 6 to get back to +10. -6 is the value of b giving us:
(x + 4)2 - 6
The question isn't over yet though, you need to solve it! Firstly equate the equation to 0:
(x + 4)2 - 6 = 0
(x + 4)2 = 6 Add +6 to both sides
x + 4 = Square root both sides
x = - 4 And then subtract 4 from both sides
The equation has been solved, the two roots are x = - 4 and x =
- 4
The natural expression is what do you do when the coefficient of x2 isn't 1? You simply factor out (put into brackets) the coefficient of x2 and repeat the same process. For example if you had to solve 3x2 + 12x + 27 you would take the coefficient of x2, here it is 3, outside the expression to give 3(x2 + 4x + 9). You then complete the square using the expression inside the brackets to give you 3((x - 2)2 + 5). You can then multiple the terms in the brackets by 3 to give 3(x - 2)2 + 15 and then rearrange to solve x.
Turning point and line of symmetry
An important fact to remember about completeing the square that the graph of (x + a)2 + b has a turning point at (-a, b) and a line of symmetry x = -a.
Solution by factorisation
To solve by factorisation you want to put the quadratic equation into brackets. You want to find two numbers that multiply together to get the constant and also add to get the coefficient of x.
Question: Solve x2 + 6x + 8
Worked solution: To start find two numbers that multiply to give 8 and also add to give 6. These would be 2 and 4, therefore the equation becomes: (x + 2)(x + 4).
For (x + 2)(x + 4) to equal 0 either bracket or both must equal 0, therefore: x + 2 = 0 or x + 4 = 0. Rearranging shows x is either -2 or -4
Quadratic formula
The quadratic formula can be used to solve equations that you wouldn't be able to factorise. First of you need to know that the general form of an quadratic equation is: ax2 + bx + c. So for example in the equation x2 + 4x + 1, a = 1, b = 4 and c = 1.
The formula
The quadratic formula is as follows:
It looks rather complex but it is simply a case of substituting in the values for a, b and c and working it out. Using the example above where a = 1, b = 4 and c = 1:
Therefore the solutions to the equation are x = and x =
The discriminant
Since you can't square root a negative number the most important part of the quadratic formula is as its value decides on the solutions to the equation. If the discriminant is less than 0 then the equation has no real roots, if it is 0 exactly then it has a single repeated root and if it is greater than 0 it has two real roots.