Completing the Square: Solve Quadratic Equations
Are you preparing for O-level exams?
Here is a quick crash course on solving Quadratic Equations by completing the square.
Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
What is Completing the Square?
Completing the square is a mathematical technique used to solve quadratic equations and rewrite quadratic expressions in a more convenient form. This method involves manipulating the equation to create a perfect square trinomial, which can then be factored or solved using the square root property. By completing the square, we transform a quadratic equation into a form that is easier to work with, making it a powerful tool for solving quadratic equations. This technique is widely used in algebra, calculus, and other areas of mathematics, providing a systematic approach to finding solutions.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. Quadratic equations have the general form ax^2 + bx + c = 0, where a, b, and c are constants. These equations can be solved using various methods, including factoring, the quadratic formula, and completing the square. Among these methods, completing the square is particularly popular because it allows us to rewrite the equation in a more convenient form, making it easier to solve for the variable. Understanding the structure of quadratic equations is essential for mastering the techniques used to solve them.
Solving Quadratic Equations using Completing the Square
Completing the square is a powerful method for solving quadratic equations. To solve a quadratic equation using completing the square, follow these steps:
- Move the constant term to the right side of the equation.
- Take half of the coefficient of the x term and square it.
- Add the result to both sides of the equation.
- Rewrite the left side of the equation as a perfect square trinomial.
- Take the square root of both sides.
- Solve for the variable.
By following these steps, you can transform a quadratic equation into a form that is easier to solve, allowing you to find the solutions more efficiently.
Examples:
To solve the quadratic equation ax2 + bx + c = 0 by completing the square, you can follow the steps below:
Step 1: Change coefficient of x2 equal to 1
a( x2 + bx/a + c/a) = 0
Step 2: Leave x2 and x terms on the Left Hand Side (LHS)
x2 + bx/a = – c/a
Step 3: Coefficient of x ÷2, square it, add to both sides
x2 + bx/a + (b/2a)2 = – c/a + (b/2a)2
Step 4: Factorise the LHS of the equation
(x + b/2a)2 = – c/a + (b/2a)2
Step 5: Simplify to solve for x
(x + b/2a)2 = – c/a + (b/2a)2
= –4ac/4a2 + b2/4a2
= (b2 – 4ac)/4a2
x + b/2a = ±√( (b2 – 4ac)/4a2 )
x = – b/2a ±√( (b2 – 4ac)/4a2 )
x =( – b ±√( (b2 – 4ac) ) /2a
Does the answer look familiar to you?
Yes, it is the Quadratic Formula!
In fact, the answer to the “Completed Square” equation is the same as the Quadratic Formula.
But you are required to show the steps to completing the square.
Let’s look at an example below.
Example (Solve Quadratic Equations by Completing the Square)
2x2 – 3x – 9 = 0
Step 1: Change coefficient of x2 equal to 1
x2 – 3x/2 – 9/2 = 0
Step 2: Leave x2 and x terms on the Left Hand Side (LHS)
x2 – 3x/2 = 9/2
Step 3: Coefficient of x ÷2, square it, add to both sides
x2 – 3x/2 + (– 3/4)2 = 9/2 + (– 3/4)2
Step 4: Factorise the LHS of the equation
(x – 3/4)2 = 9/2 + 9/16
= 81/16
Step 5: Simplify to solve for x
(x – 3/4)2 = ± 9/4
x = 3/4 + 9/4 or x = 3/4 – 9/4
= 3 = –3/2 (ans)
Last Minute Revision for O Level Math?
Check out our exam guide on other topics here!
Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
Does your child need help with Mathematics? Find out more about our Math Tuition Class Free Trial.
1) Live Zoom Lessons at Grade Solution Learning Centre
At Grade Solution Learning Centre, we are a team of dedicated educators whose mission is to guide your child to academic success. Here are the services we provide:
– Live Zoom lessons
– EdaptIQ™, a smart learning platform that tracks your child’s progress, strengths and weaknesses through personalised digital worksheets.
– 24/7 Homework Helper Service
We provide all these services above at a very affordable monthly fee to allow as many students as possible to access such learning opportunities. We also offer a free trial class and consultation with our tutors, to help make sure we are the best fit for your child.
We specialise in English, Math, and Science subjects.
You can see our fees and schedules here >>
2) Pre-recorded Online courses on Jimmymaths.com
If you are looking for something that fits your budget, or prefer your child learn at his or her own pace, you can join our pre-recorded online Math courses.
Your child can:
– Learn from recorded videos
– Get access to lots of common exam questions to ensure sufficient practice
– Get unlimited support and homework help
You can see the available courses here >>