A quadratic equation is an equation in which the highest power of the variable is 2. You will come across quadratic equations in algebra, graphs, measurements, word problems, and secondary school mathematics.
The good news is that most quadratic equations can be solved using a few clear methods. The main ones are factorisation, the quadratic formula, completing the square, and using a graph.
In this helpful post, you will learn how to identify a quadratic equation, choose the right method, solve different types of quadratic equations step by step, and check whether your answer is correct.
What is a quadratic equation?
A quadratic equation is usually written in this form
ax² + bx + c = 0
where a, b, and c are numbers, and a cannot be zero.
For example
x² + 5x + 6 = 0
2x² – 7x + 3 = 0
3x² + 4x – 8 = 0
All three are quadratic equations because the highest power of x is 2.
The value of a is the coefficient of x², b is the coefficient of x, and c is the constant.
For
2x² – 7x + 3 = 0
we have
a = 2
b = -7
c = 3
What does it mean to solve a quadratic equation?
Solving a quadratic equation means finding the value or values of x that make the equation true.
For example
x² – 5x + 6 = 0
There are two values of x that make this equation true.
They are
x = 2
and
x = 3
We can check them.
If x = 2
2² – 5(2) + 6 = 0
4 – 10 + 6 = 0
So x = 2 works.
If x = 3
3² – 5(3) + 6 = 0
9 – 15 + 6 = 0
So x = 3 also works.
Therefore, the solutions are x = 2 and x = 3.
Before solving, put the equation equal to zero
One of the most important steps in solving a quadratic equation is to arrange it in the standard form
ax² + bx + c = 0
For example
x² + 7x = 18
This is not yet in standard form because the right side is not zero.
Move 18 to the left
x² + 7x – 18 = 0
Now it is ready to solve.
This small step prevents many mistakes, especially when using the quadratic formula.
Method 1 Solving quadratic equations by factorisation
Factorisation is often the quickest method when the quadratic expression can be easily factorised.
Consider
x² + 5x + 6 = 0
We need to find two numbers that
- multiply to give 6
- add to give 5
The two numbers are 2 and 3.
So we can write
x² + 5x + 6 = (x + 2)(x + 3)
Therefore
(x + 2)(x + 3) = 0
For the product of two expressions to equal zero, at least one of the expressions must be zero.
So
x + 2 = 0
or
x + 3 = 0
Solve each one
x = -2
or
x = -3
Therefore
x = -2 or x = -3
Another example using factorisation
Solve
x² – 7x + 12 = 0
We need two numbers that multiply to give 12 and add to give -7.
The numbers are -3 and -4.
Therefore
x² – 7x + 12 = (x – 3)(x – 4)
So
(x – 3)(x – 4) = 0
Therefore
x – 3 = 0
or
x – 4 = 0
So
x = 3
or
x = 4
The answer is
x = 3 or x = 4
How to know which numbers to use when factorising
This is where many students get stuck.
For an equation such as
x² + 7x + 12 = 0
look for two numbers whose product is 12 and whose sum is 7.
Possible factor pairs of 12 include
1 and 12
2 and 6
3 and 4
Since 3 + 4 = 7, we use 3 and 4.
Therefore
x² + 7x + 12 = (x + 3)(x + 4)
What if the constant is negative?
Consider
x² + x – 6 = 0
We need two numbers that multiply to -6 and add to 1.
The numbers are 3 and -2.
Therefore
x² + x – 6 = (x + 3)(x – 2)
So
(x + 3)(x – 2) = 0
Therefore
x = -3
or
x = 2
A useful way to think about this is that a negative product means the two numbers must have opposite signs.
Method 2 Solving quadratic equations using the quadratic formula
Not every quadratic equation factorises easily.
This is where the quadratic formula becomes extremely useful.
For
ax² + bx + c = 0
the quadratic formula is
x = (-b ± √(b² – 4ac)) / 2a
The ± symbol means that you use the formula twice.
Once with a plus sign and once with a minus sign.
Example using the quadratic formula
Solve
x² + 3x – 4 = 0
First identify the values of a, b, and c.
a = 1
b = 3
c = -4
Substitute them into the formula
x = (-3 ± √(3² – 4(1)(-4))) / 2(1)
Work inside the square root first
x = (-3 ± √(9 + 16)) / 2
x = (-3 ± √25) / 2
Since √25 = 5
x = (-3 ± 5) / 2
Now calculate the two answers.
Using the plus sign
x = (-3 + 5) / 2
x = 2 / 2
x = 1
Using the minus sign
x = (-3 – 5) / 2
x = -8 / 2
x = -4
Therefore
x = 1 or x = -4
A quadratic equation where factorisation is difficult
Consider
2x² + 3x – 7 = 0
This does not factorise neatly using simple whole numbers, so the quadratic formula is a good choice.
Here
a = 2
b = 3
c = -7
Substitute into the formula
x = (-3 ± √(3² – 4(2)(-7))) / 4
x = (-3 ± √(9 + 56)) / 4
x = (-3 ± √65) / 4
Since √65 cannot be simplified to a whole number, the exact answers are
x = (-3 + √65) / 4
and
x = (-3 – √65) / 4
This is completely acceptable as an exact answer.
If a decimal answer is required, you can use a calculator to obtain approximate values.
Method 3 Solving quadratic equations by completing the square
Completing the square is another important method for solving quadratic equations.
It is especially useful when you want to understand how quadratic equations connect to the vertex of a parabola.
Consider
x² + 6x + 5 = 0
Move the constant to the other side
x² + 6x = -5
Take half of the coefficient of x.
The coefficient is 6.
Half of 6 is 3.
Square 3
3² = 9
Add 9 to both sides
x² + 6x + 9 = -5 + 9
The left side is now a perfect square
(x + 3)² = 4
Take the square root of both sides
x + 3 = ±2
Therefore
x = -3 ± 2
So
x = -1
or
x = -5
Therefore the solutions are
x = -1 or x = -5
Why do we use the plus-or-minus sign?
Suppose
x² = 25
There are two possible values of x
x = 5
and
x = -5
because
5² = 25
and
(-5)² = 25
That is why solving a squared equation normally gives a positive and negative possibility.
Forgetting the negative answer is a common mistake when completing the square.
Method 4 Solving a quadratic equation using a graph
A quadratic equation can also be understood graphically.
If you have
y = x² – 5x + 6
draw the graph of the equation.
The solutions of
x² – 5x + 6 = 0
are the points where the graph crosses the x-axis.
For this example, the graph crosses the x-axis at
x = 2
and
x = 3
The graphical method is useful for understanding what the solutions mean, although factorisation or the quadratic formula is usually more practical when an exact answer is required.
How to choose the best method
You do not always need to use the quadratic formula.
A good approach is to look at the equation first.
Use factorisation when
The numbers are easy to work with and the quadratic factorises neatly.
Example
x² – 9x + 20 = 0
This can be factorised as
(x – 4)(x – 5) = 0
So factorisation is much faster than using the quadratic formula.
Use the quadratic formula when
The equation does not factorise easily or you are unsure how to factorise it.
For example
3x² + 2x – 7 = 0
The quadratic formula will always work for a quadratic equation where a ≠ 0.
Use completing the square when
You need to show the process of completing the square, or you want to understand the structure of the quadratic.
Use a graph when
You want to see the solutions visually or estimate the solutions.
How to solve a quadratic equation with a coefficient in front of x²
Some students are comfortable with equations such as
x² + 5x + 6 = 0
but become confused when the coefficient of x² is greater than 1.
Consider
2x² + 7x + 3 = 0
We need to factorise it.
Multiply the coefficient of x² by the constant
2 × 3 = 6
Now find two numbers that multiply to 6 and add to 7.
They are 6 and 1.
Split the middle term
2x² + 6x + x + 3 = 0
Group the terms
2x(x + 3) + 1(x + 3) = 0
Factorise
(2x + 1)(x + 3) = 0
Therefore
2x + 1 = 0
or
x + 3 = 0
So
x = -1/2
or
x = -3
This method is sometimes called splitting the middle term.
How to solve quadratic equations with negative coefficients
Negative signs can make quadratic equations look more complicated than they really are.
Consider
-x² + 5x – 6 = 0
It is often easier to multiply the entire equation by -1.
This gives
x² – 5x + 6 = 0
Now factorise
(x – 2)(x – 3) = 0
Therefore
x = 2
or
x = 3
Multiplying the whole equation by -1 does not change its solutions.
How to solve quadratic equations with fractions
Fractions can make an equation harder to read, but you can usually remove them first.
Consider
x²/2 + x/2 – 1 = 0
Multiply every term by 2
x² + x – 2 = 0
Now factorise
(x + 2)(x – 1) = 0
Therefore
x = -2
or
x = 1
Removing fractions at the beginning can make the rest of the calculation much easier.
How to check your quadratic equation answer
Always check your answers when possible.
Suppose you solved
x² – 5x + 6 = 0
and obtained
x = 2 and x = 3
Put each value back into the original equation.
For x = 2
2² – 5(2) + 6 = 0
4 – 10 + 6 = 0
For x = 3
3² – 5(3) + 6 = 0
9 – 15 + 6 = 0
Both values work, so the answers are correct.
Checking is particularly important in examinations because one small sign error can change the final answer.
The discriminant and what it tells you
When using the quadratic formula, you will see the expression
b² – 4ac
This is called the discriminant.
It can tell you how many real solutions a quadratic equation has.
If b² – 4ac is positive
There are two different real solutions.
For example
x² – 5x + 6 = 0
The discriminant is
(-5)² – 4(1)(6)
25 – 24 = 1
Since 1 is positive, there are two different real solutions.
If b² – 4ac is zero
There is one repeated real solution.
For example
x² – 6x + 9 = 0
The discriminant is
(-6)² – 4(1)(9)
36 – 36 = 0
So there is one repeated solution.
In fact
x² – 6x + 9 = (x – 3)²
so the solution is
x = 3
If b² – 4ac is negative
There are no real-number solutions.
For example
x² + 2x + 5 = 0
The discriminant is
2² – 4(1)(5)
4 – 20 = -16
Because the discriminant is negative, the equation has no real solutions.
Common mistakes when solving quadratic equations
Forgetting to make the equation equal to zero
The quadratic formula is designed for an equation in the form
ax² + bx + c = 0
Do not substitute values into the formula before rearranging the equation.
Getting the sign of b wrong
If the equation is
x² – 7x + 10 = 0
then
b = -7
It is not +7.
This matters because the quadratic formula contains -b.
Forgetting the ± sign
The quadratic formula is
x = (-b ± √(b² – 4ac)) / 2a
The plus and minus signs normally produce two possible solutions.
Taking the square root incorrectly
Remember that
√25 = 5
but when solving
x² = 25
the solutions are
x = 5 or x = -5
The negative possibility must not be forgotten.
Expanding brackets incorrectly
For example
(x + 3)(x + 2)
expands to
x² + 5x + 6
because
x × x = x²
x × 2 = 2x
3 × x = 3x
3 × 2 = 6
So
x² + 2x + 3x + 6 = x² + 5x + 6
Careful expansion makes factorisation much easier to check.
Quadratic equations in word problems
Quadratic equations are not only found in algebra exercises. They can also appear in real-life problems involving areas, dimensions, numbers, and measurements.
For example, suppose a rectangular garden has a width of x metres and a length of x + 3 metres. If its area is 40 square metres, we can write
x(x + 3) = 40
Expand
x² + 3x = 40
Move everything to one side
x² + 3x – 40 = 0
Factorise
(x + 8)(x – 5) = 0
Therefore
x = -8
or
x = 5
Because a physical length cannot be negative, we reject -8.
So the width is
5 metres
and the length is
8 metres.
This is an important lesson in word problems. Mathematics may give you more than one algebraic solution, but the situation in the question determines which answer makes sense.
Quadratic equations for WAEC and other mathematics examinations
Quadratic equations are an important part of secondary school algebra, so it is worth learning the method rather than memorising one example.
When facing a quadratic equation in an examination, use this process
Step 1 Rearrange the equation so one side is zero.
Step 2 Identify the values of a, b, and c if you are using the quadratic formula.
Step 3 Look at the equation and decide whether factorisation is easy.
Step 4 Factorise or apply the quadratic formula carefully.
Step 5 Find all possible values of x.
Step 6 Check your answers.
Step 7 If the question is a word problem, make sure the answer makes sense in the real situation.
Do not rush through the signs. A large number of quadratic equation mistakes come from simple addition, subtraction, or negative-sign errors rather than from the main method itself.
Practice questions on quadratic equations
Try these questions before looking at the answers.
Question 1
Solve
x² + 7x + 12 = 0
Question 2
Solve
x² – 9x + 20 = 0
Question 3
Solve
x² – 2x – 15 = 0
Question 4
Solve
2x² + 7x + 3 = 0
Question 5
Solve
x² + 4x + 1 = 0
Question 6
Solve
3x² – 5x – 2 = 0
Question 7
Solve
x² – 10x + 25 = 0
Question 8
Solve
2x² + x – 6 = 0
Answers
1. x = -3 or x = -4
2. x = 4 or x = 5
3. x = 5 or x = -3
4. x = -1/2 or x = -3
5. x = -2 + √3 or x = -2 – √3
6. x = 2 or x = -1/3
7. x = 5
8. x = 3/2 or x = -2
Frequently asked questions about quadratic equations
What is the easiest way to solve a quadratic equation?
If the equation factorises easily, factorisation is usually the quickest method. If it does not factorise neatly, the quadratic formula is a reliable choice.
What are the four methods of solving quadratic equations?
The main methods are factorisation, the quadratic formula, completing the square, and graphical methods.
Can every quadratic equation be solved by factorisation?
No. Some quadratic equations do not factorise neatly using simple integers. The quadratic formula can be used in such cases.
What is the quadratic formula?
For an equation in the form ax² + bx + c = 0, the quadratic formula is
x = (-b ± √(b² – 4ac)) / 2a
Why does a quadratic equation sometimes have two answers?
A quadratic equation can have two different real solutions because a parabola can cross the x-axis at two different points.
Can a quadratic equation have only one answer?
Yes. If the discriminant is zero, the equation has one repeated real solution.
Can a quadratic equation have no real answer?
Yes. If the discriminant is negative, there are no real-number solutions.
How do I know which method to use?
First check whether the equation factorises easily. If it does, factorisation is usually convenient. If factorisation is difficult, use the quadratic formula. Completing the square is another useful method, especially when the question specifically asks for it.
Quick summary
To solve a quadratic equation
1. Put the equation in the form ax² + bx + c = 0.
2. Try factorisation if the numbers are simple.
3. If factorisation is difficult, use the quadratic formula.
4. You can also use completing the square or a graph.
5. Remember the ± sign when using square roots or the quadratic formula.
6. Check your solutions in the original equation.
The most important thing is not to memorise one example. Learn why each step works, and you will be able to solve many different quadratic equations even when the numbers change.
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