When should you solve a quadratic equation by factorisation ?

You should solve a quadratic equation by factorisation when: the question asks you to solve by factorisation, solving two-term quadratic equations, a calculator gives solutions as whole numbers or fractions (indicating that the quadratic factorises).

When should you use the quadratic formula to solve a quadratic equation?

You should use the quadratic formula to solve a quadratic equation when: the question asks to give solutions correct to a given accuracy, e.g. 2 decimal places, you think the quadratic formula might be faster than factorising, in doubt (because the quadratic formula always works!)

True or False? Calculators can be used to check final solutions when solving quadratic equations.

True. Calculators can be used to check final solutions when solving quadratic equations. Make sure that you know how to use your particular calculator model to solve quadratic equations.

What does it tell you if a calculator gives solutions to a quadratic equation as whole numbers or fractions ?

If a calculator gives solutions to a quadratic equation as whole numbers or fractions (with no square roots), it tells you that the quadratic equation can be factorised .

IGCSEMathsCambridge (CIE)ExtendedFlashcards

Quadratic Equations (Cambridge (CIE) IGCSE Maths)

Flashcards

1/23

FrontSolving Quadratics by Factorising
What is a quadratic equation?
FrontSolving Quadratics by Factorising
What does it mean to solve a quadratic equation by factorising?
BackSolving Quadratics by Factorising
Solving a quadratic equation by factorising means:
Making sure it is in the form $a x^{2} + b x + c = 0$ (i.e. with zero on one side).
Factorising the quadratic.
Setting each bracket equal to zero to find the solutions.

Enjoying Flashcards?
Tell us what you think

Cards in this collection (23)

What is a quadratic equation?
A quadratic equation is an equation of the form $a x^{2} + b x + c = 0$ ,
where $a$ , $b$ , and $c$ are constants, and $a$ is not equal to zero.
What does it mean to solve a quadratic equation by factorising?
Solving a quadratic equation by factorising means:
1. Making sure it is in the form $a x^{2} + b x + c = 0$ (i.e. with zero on one side).
2. Factorising the quadratic.
3. Setting each bracket equal to zero to find the solutions.
True or False?
If $(x + 4) (x - 1) = 0$ , then either $x + 4 = 0$ or $x - 1 = 0$ .
True.
If $(x + 4) (x - 1) = 0$ , then either $x + 4 = 0$ or $x - 1 = 0$ .
True or false?
The solutions of the equation $(x - 2) (x + 5) = 0$ are $x = - 2$ and $x = 5$ .
False.
To find the solutions of the equation $(x - 2) (x + 5) = 0$ , you should solve the linear equations $x - 2 = 0$ and $x + 5 = 0$ .
The solutions are $x = 2$ and $x = - 5$ .
Note that the signs in front of 2 and 5 are the 'other way round' to how they appear in the brackets.
What two linear equations should you solve to find the solutions of the quadratic equation $(8 x + 7) (2 x - 3) = 0$ ?
To find the solutions of the quadratic equation $(8 x + 7) (2 x - 3) = 0$ , you should solve the linear equations $8 x + 7 = 0$ and $2 x - 3 = 0$ .
The solutions are $x = - \frac{7}{8}$ and $x = \frac{3}{2}$ .
True or false?
$x = 0$ is one of the solutions of the quadratic equation $x (5 x - 1) = 0$ .
True.
If $x (5 x - 1) = 0$ , then either $x = 0$ or $5 x - 1 = 0$ .
So $x = 0$ is one of the solutions.
The other solution is $x = \frac{1}{5}$ .
What is the quadratic formula?
The quadratic formula can be used to solve a quadratic equation of the form $a x^{2} + b x + c = 0$ .
The quadratic formula gives the solutions as $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .
The quadratic formula is on the exam formula sheet, so you don't need to memorise it. But you do need to be able to use it.
True or False?
You need ' $= 0$ ' on one side of a quadratic equation to use the quadratic formula.
True.
You need ' $= 0$ ' on one side of a quadratic equation to use the quadratic formula.
What does it mean if $b^{2} - 4 a c$ is negative when using the quadratic formula?
If $b^{2} - 4 a c$ is negative when using the quadratic formula, it means there are no real solutions to the quadratic equation.
When completing the square for the expression $x^{2} + b x + c$ , explain how to find the value of $p$ in the expression ${(x + p)}^{2} + q$ .
Completing the square for $x^{2} + b x + c$ gives the form ${(x + p)}^{2} + q$ .
The value of $p$ is half of the value of $b$ .
If an equation is given in completed-square form, such as ${(x + 3)}^{2} - 4 = 0$ , explain how to solve it.
If an equation is given in completed-square form, such as ${(x + 3)}^{2} - 4 = 0$ , to solve it you need to make $x$ the subject.
Add 4 to both sides: ${(x + 3)}^{2} = 4$
Take $\pm$ square roots: $x + 3 = \pm 2$
Make $x$ the subject: $x = \pm 2 - 3$ giving $x = - 1$ or $x = - 5$
(Note that to solve it, you do not expand the brackets back out!)
True or False?
The coordinates of the turning point (vertex) of the quadratic curve $y = {(x + 3)}^{2} + 5$ are $(3, 5)$ .
False.
The coordinates of the turning point (vertex) of the quadratic curve $y = {(x + 3)}^{2} + 5$ are not $(3, 5)$ .
The correct coordinates are $(- 3, 5)$ .
This is a common mistake in the exam! If $y = {(x + p)}^{2} + q$ is the curve, then $(- p, q)$ are the coordinates of the turning point.
What is the first step to completing the square of the quadratic expression $a x^{2} + b x + c$ where $a$ is not equal to 1, e.g. $3 x^{2} + 12 x - 8$ ?
The first step to completing the square of the quadratic expression $a x^{2} + b x + c$ where $a$ is not equal to 1 is to factorise out $a$ .
This gives $a [x^{2} + \frac{b}{a} x + \frac{c}{a}]$ .
You can then complete the square inside the big brackets.
E.g. To complete the square of the expression $3 x^{2} + 12 x - 8$ , the first step would be to factorise out $3$ , giving $3 [x^{2} + 4 + \frac{8}{3}]$ .
True or False?
The coordinates of the turning points on the curves $y = {(x + 2)}^{2} + 4$ and $y = 3 {(x + 2)}^{2} + 4$ are the same.
True.
The coordinates of the turning points on the curves $y = {(x + 2)}^{2} + 4$ and $y = 3 {(x + 2)}^{2} + 4$ are the same.
The coordinates of the turning point on $y = a {(x + p)}^{2} + q$ are always $(- p, q)$ , regardless of the value of $a$ (even if $a < 0$ ).
Explain how writing $x^{2} - 4 x + 9$ in the form ${(x - 2)}^{2} + 5$ shows that any output of the function $f (x) = x^{2} - 4 x + 9$ is always greater than, or equal to, 5.
If $x^{2} - 4 x + 9$ can be written as ${(x - 2)}^{2} + 5$ by completing the square, then $f (x) = x^{2} - 4 x + 9$ can be written as $f (x) = {(x - 2)}^{2} + 5$ .
The coordinates of the turning point will be $(2, 5)$ .
As this is a positive quadratic, the turning point will be a minimum and the therefore the value of $f (x)$ will always be greater than or equal to 5.
When should you solve a quadratic equation by factorisation?
You should solve a quadratic equation by factorisation when:
- the question asks you to solve by factorisation,
- solving two-term quadratic equations,
- a calculator gives solutions as whole numbers or fractions (indicating that the quadratic factorises).
When should you use the quadratic formula to solve a quadratic equation?
You should use the quadratic formula to solve a quadratic equation when:
- the question asks to give solutions correct to a given accuracy, e.g. 2 decimal places,
- you think the quadratic formula might be faster than factorising,
- in doubt (because the quadratic formula always works!)
When should you solve a quadratic equation by completing the square?
You should solve a quadratic equation by completing the square, when:
- one part of a question says to complete the square and another part says to use the first part to solve the equation
- making $x$ the subject of harder formulae containing $x^{2}$ and
  $x$ terms
(Note that, like the quadratic formula, completing the square will always work to solve a quadratic equation. It's just not always easy to do!)
True or False?
Calculators can be used to check final solutions when solving quadratic equations.
True.
Calculators can be used to check final solutions when solving quadratic equations.
Make sure that you know how to use your particular calculator model to solve quadratic equations.
What does it tell you if a calculator gives solutions to a quadratic equation as whole numbers or fractions?
If a calculator gives solutions to a quadratic equation as whole numbers or fractions (with no square roots), it tells you that the quadratic equation can be factorised.
True or False?
When solving a quadratic equation of the form $x (a x + b) = 0$ , the first step is to divide both sides by $x$ .
False.
It is a common mistake to divide both sides by $x$ at the beginning when solving a quadratic equation of the form $x (a x + b) = 0$
It is a mistake because if you do this you will lose the $x = 0$ solution.
Instead, solve $x = 0$ and $a x + b = 0$ .
True or False?
When solving a quadratic equation of the form $a x^{2} + b x + c = 0$ by completing the square, the first step should be to divide both sides by $a$ .
True.
When solving a quadratic equation of the form $a x^{2} + b x + c = 0$ by completing the square, you should divide both sides by $a$ first.
True or False?
The quadratic formula actually comes from completing the square to solve $a x^{2} + b x + c = 0$ .
True.
The quadratic formula actually comes from completing the square to solve $a x^{2} + b x + c = 0$ .
(You don't need to be able to do this on the exam. But doing this is a great way to practise your algebra skills!)

Number

Algebra & Sequences

Coordinate Geometry & Graphs

Geometry

Lengths, Areas & Volumes

Pythagoras & Trigonometry

Vectors & Transformations

Probability

Statistics