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Solving a Quadratic Equation
(KS4, Year 10)
The Lesson
A quadratic equation is an equation in the form:Solving a quadratic equation mean finding the value of x that makes this equation true (i.e. makes the left hand side equal to 0.) The values of x that solve the equation are called the roots of the equation.
Understanding Solving Quadratic Equations
It is easier to understand solving quadratic equations with an example. Let's look at a quadratic equation.x is a variable. It can take different values. Let's try x = 1, x = 2 and x = 3.
x = 1
Substitute x = 1 into the left hand side of the quadratic equation:x^{2} − 3x + 2 = (1 )^{2} − 3(1 ) + 2
x^{2} − 3x + 2 = 1 × 1 − 3 × 1 + 2
x^{2} − 3x + 2 = 1 − 3 + 2
x^{2} − 3x + 2 = 0
x = 2
Substitute x = 2 into the left hand side of the quadratic equation:x^{2} − 3x + 2 = (2 )^{2} − 3(2 ) + 2
x^{2} − 3x + 2 = 2 × 2 − 3 × 2 + 2
x^{2} − 3x + 2 = 4 − 6 + 2
x^{2} − 3x + 2 = 0
x = 3
Substitute x = 3 into the left hand side of the quadratic equation:x^{2} − 3x + 2 = (3 )^{2} − 3(3 ) + 2
x^{2} − 3x + 2 = 3 × 3 − 3 × 3 + 2
x^{2} − 3x + 2 = 9 − 9 + 2
x^{2} − 3x + 2 = 2 ≠ 0
How to Solve Quadratic Equations
There are 3 ways to solve quadratic equations.(1) Factoring
A quadratic equation can sometimes be written as the product of two brackets. For example:from this, we can read off the two roots of the quadratic equation: x = 1, x = 2
Read more about solving quadratic equations using factoring
(2) Quadratic Formula
A quadratic equation can be solved using the quadratic formula:In this formula, a, b and c are the numbers in the quadratic equation in standard form, ax^{2} + bx + c.
Read more about solving quadratic equations using the quadratic formula
(3) Graph
A quadratic equation can be solved by plotting it on a graph and finding where it crosses the xaxis:In this graph above, the quadratic curve crosses the xaxis at x = 1 and x = 2. These are the roots of the equation, that solve the equation.
Interactive Widget
You can use this interactive widget to create a graph of your quadratic equation. Use the buttons to change the values of the quadratic equation.Read more about solving quadratic equations using a graph
What's in a Name?
The word "quadratic" comes from the word "quad", meaning "square"  because the x is squared.Factoring, Factorising
To write a quadratic equation as a product of two brackets is called 'to factor' or 'to factorise' the quadratic equation. The method is refered to as 'factoring' or 'factorising'.There Are 2 Roots
Quadratic equations always have two solutions. There are 2 values of x that solve the equation. We can visualize this by looking at a graph of a quadratic equation. The roots are the points where the curve crosses the horizontal xaxis.
There can be two distinct roots. We see this because the curve crosses the xaxis at 2 separate places:

Sometimes it seems that there is only one root. But that root is repeated.

Even when it seems there are no roots, there are two complex roots.
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