Equations » Quadratic equations (x2 = number / sum-product-method / the quadratic formula)
Quadratic equations can be divided into three categories.
Every category has its own way of solving the equation.
|Reducible to x2 = number||x2 = 25 or x2 – 7 = 2|
|Reducible to binomial = 0||5x2 – x = 0 or 3x2 + 5x = x2 – 7x|
|Reducible to trinomial = 0||5x2 – 4x + 3 = 0 or 3x2 = 2x2 + 4x + 12|
Directly to other theory within this page:– Factorising formulas (incl. sum-product-method)
– The quadratic formula / abc-formula
– Completing the square
– Calculate the vertex
– Shape of the parabola
– Calculate the points of intersection with the axes
– Quadratic equations with a parameter
– More examples (mixed)
x2 = number
Binomial = 0
Reduce to zero and factorise the binomial (single brackets). After factorising, either the left-hand factor or the right-hand factor must be zero.
Trinomial = 0
Reduce to zero and then factorise the trinomial (double brackets). When factorising is not possible, use the quadratic formula / abc-formula or completing the square.
5x2 – 4x – 3 = 0
a = 5, b = –4, c = –3
D = (–4)2 – 4 · 5 · –3 = 76
Single brackets (binomial)
Look for the greatest common divider in both terms.
That is what you write in front of the brackets.
Your factorised formula is always of the form y = …(… + …) or y = …(… – …).
Example: In both terms of y = 6x2 + 15x you have a factor 3x
After all, the formula can be written as y = 2 × 3 × x × x + 3 × 5 × x.
This formula can therefore be factorised and written as y = 3x(2x + 5).
A sum is the answer of an addition and a product is the answer of a multiplication.
The sum-product-method can only be used with a trinomial that is
written as x2 + bx + c = 0 (so a = 1).
Underneath this trinomial you write double brackets like this: (x … …)(x … …) = 0
Behind the x on the open spaces you have to fill in the two numbers that are added b and multiplied c. Write a + in front of positive numbers.
15x2 + 15x – 6 = 0
get a = 1 by multiplying every term by 5
x2 + x – 30 = 0
(x – 5)(x + 6) = 0
x = 5 or x = –6
5x2 + 25x – 70 = 0
get a = 1 by dividing every term by 5
x2 + 5x – 14 = 0
(x – 2)(x + 7) = 0
x = 2 or x = –7
Check factorising when you want to learn more about factorising.
The quadratic formula / abc-formula
This formula can be used to solve every quadratic equation.
However, only use it on trinomials that cannot be factorised, as it costs a lot of work.
The equation has to be written as ax2 + bx + c = 0.
Find the numbers for a, b and c. Watch out with negative numbers!
With these numbers you first calculate the Discriminant, the formula is:
D = b2 – 4ac
When D is negative, you have no solutions.
When D is zero, you have one solution.
When D is positive, you have two solutions.
When D is not negative, use the following two formulas to calculate the values of x.
x = –b + 2a or x = –b – 2a
(–b + ) / (2 × a)
Do you want to know how this formula is derived?
Look at derivation of the quadratic formula.
Example 2 at 'Trinomial = 0' above and example 6 all the way at the bottom of the page are examples of the quadratic formula.
Completing the square
You can also solve a quadratic equation by completing the square. However, it is not necessary to learn this. After all, you can solve any quadtratic equation with the quadratic formula / abc-formula. Look at completing the square for the theory and a number of examples on how you can solve a quadratic equation by completing the square.
Calculate the vertex
Use xvertex = –b2a.
To calculate yvertexyou have to fill in xvertexinto the formula.
Check vertices of parabolas for more information about calculating the vertex.
Shape of the parabola
Is completely determined by a.
When a = positive you have an upward-opening parabola
When a = negative you have a downward-opening parabola
The bigger the difference between a and zero, the narrower the parabola is.
Calculate the points of intersection with the axes
For the intersection with the y-axis, calculate y for x = 0.
For the intersections with the x-axis calculate x for y = 0.
Quadratic equations with a parameter
When you have an equation like 2x2 + 5x + p = 0 or –x2 + 2x + 1,5 = –4x + p, check quadratic equations with a parameter.
More examples (mixed)
D = (–12)2 – 4 × 3 × 9.5 = 30