Solving a Quadratic Equation
(KS4, Year 10)
The LessonA 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 EquationsIt 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 = 1Substitute x = 1 into the left hand side of the quadratic equation:
x2 − 3x + 2 = (1 )2 − 3(1 ) + 2
x2 − 3x + 2 = 1 × 1 − 3 × 1 + 2
x2 − 3x + 2 = 1 − 3 + 2
x2 − 3x + 2 = 0
x = 2Substitute x = 2 into the left hand side of the quadratic equation:
x2 − 3x + 2 = (2 )2 − 3(2 ) + 2
x2 − 3x + 2 = 2 × 2 − 3 × 2 + 2
x2 − 3x + 2 = 4 − 6 + 2
x2 − 3x + 2 = 0
x = 3Substitute x = 3 into the left hand side of the quadratic equation:
x2 − 3x + 2 = (3 )2 − 3(3 ) + 2
x2 − 3x + 2 = 3 × 3 − 3 × 3 + 2
x2 − 3x + 2 = 9 − 9 + 2
x2 − 3x + 2 = 2 ≠ 0
How to Solve Quadratic EquationsThere are 3 ways to solve quadratic equations.
(1) FactoringA 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 FormulaA 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, ax2 + bx + c.
Read more about solving quadratic equations using the quadratic formula
(3) GraphA quadratic equation can be solved by plotting it on a graph and finding where it crosses the x-axis:
In this graph above, the quadratic curve crosses the x-axis at x = 1 and x = 2. These are the roots of the equation, that solve the equation.
Interactive WidgetYou 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, FactorisingTo 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 RootsQuadratic 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 x-axis.
There can be two distinct roots. We see this because the curve crosses the x-axis 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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