Download Key Introduction What is a Quadratic Equation?

Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
Key
On screen content
Narration – voice-over
Activity – Under the Activities heading of the online program
Introduction
This topic will cover:
• the definition of a quadratic equation;
• how to solve a quadratic equation when b = 0;
• how to solve a quadratic equation by factorising; and
• how to solve a quadratic equation using the quadratic formula.
Welcome to Algebra.
This topic will cover:
•
•
•
•
the definition of a quadratic equation;
how to solve a quadratic equation when b = 0;
how to solve a quadratic equation by factorising; and
how to solve a quadratic equation using the quadratic formula.
What is a Quadratic Equation?
A quadratic equation, or second degree equation, is an algebraic equation of the form:
ax2 + bx + c = 0,
where x is a variable and a, b and c represent known numbers such that a ≠ 0 (if a = 0 then the
equation is linear). These are referred to as coefficients of the equation.
The most basic quadratic equation occurs when a = 1, b = 0 and c = 0, in which case we have:
x2 = 0
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
In this case you should be able to see that the value of the variable x must be 0, as 02 = 0. However in
general, determining the value of the variable (that is, solving the equation) requires a bit of working.
This topic will cover methods of solving three different types of quadratic equations.
A quadratic equation, otherwise known as a second degree equation, is an algebraic equation of the
form ax squared plus bx plus c equals zero, where x is a variable and a, b and c represent known
numbers such that a is not equal to zero. Note that if a is equal to zero then the equation is linear,
rather than quadratic. Linear equations are covered in the topic Rearranging and Solving Linear
Equations, so please refer to this if you would like more information. (if a = 0 then the equation is
linear).
The most basic quadratic equation occurs when a is equal to one and b and c are both equal to zero,
in which case the quadratic equation is simply x squared equals 0. In this case you should be able to
see that the value of the variable x must be zero, as zero is the only number that when squared is
equal to zero. However in general determining the value of the variable, or in other words, solving the
equation, requires a bit of working.
This topic will cover methods of solving three different types of quadratic equations.
Solving Quadratic Equations when b = 0
The first type of quadratic equation you should be able to solve is ax2 + bx + c = 0 when b = 0; in other
words, when the equation is of the form:
ax2 + c = 0
You can solve equations like this by following these steps:
1) Rearrange the equation so that ax2 is by itself on the left hand side of the equation, by adding or
subtracting the constant c from both sides, as applicable.
For example, if you wish to solve the quadratic equation 2x2 – 72 = 0 your first steps of working
would be:
2x2 – 72 = 0
∴2x2 – 72 + 72 = 0 + 72 (undo subtraction by adding)
∴
2x2 = 72
(simplify)
2) Divide through both sides of the equation by a (if a ≠ 1).
For example, the next steps in solving the quadratic equation 2x2 – 72 = 0 are:
∴
2x2/2 = 72/2 (undo multiplication by dividing)
∴
x2 = 36
(simplify)
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
3) Take the square root of both sides of the equation, keeping in mind that there will be both a
positive and a negative solution (except when the solution is 0); it is up to you to determine
whether or not both solutions are applicable in the context of the problem.
For example, the final steps in solving the quadratic equation 2x2 – 72 = 0 are:
∴
√x2= √36
(take square root of both sides)
∴
x = ±6
(simplify)
The first type of quadratic equation you should be able to solve is ax squared plus bx plus c equals
zero when b = 0; in other words, when the equation is of the form ax squared plus c equals zero. You
can solve equations like this by following a series of steps.
The first step is to rearrange the equation so that ax squared is by itself on the left hand side of the
equation, by adding or subtracting the constant c from both sides, as applicable. For example, if you
wish to solve the quadratic equation 2x squared subtract 72 equals zero your first steps of working
would be to undo the subtraction of 72 by adding 72 to both sides, to give 2x squared subtract 72
plus 72 equals zero plus 72. You can then simplify this equation to give 2x squared equals 72. Note
that for more information on rearranging equations in this way, please review the Rearranging and
Solving Linear Equations topic.
The second step applies to equations where a is not equal to 1. In this case, you need to divide
through both sides of the equation by a.
For example, the next step in solving the quadratic equation 2x squared subtract 72 equals zero is to
undo the multiplication of 2 by dividing both sides through by 2, and then simplifying to give x
squared equals 36.
The third and final step is to take the square root of both sides of the equation, keeping in mind that
there will be both a positive and a negative solution- except for the case when the solution is zero.
When there are two solutions, it is up to you to determine whether or not both solutions are
applicable in the context of the problem. For example, the final step in solving the quadratic equation
2x squared subtract 72 equals zero is to take the square root of both sides, which shows that x is
equal either to positive or negative 6.
Examples: Solving Quadratic Equations when b = 0
1) x2 – 64 = 0
∴x2 – 64 + 64 = 0 + 64 (undo subtraction by adding)
∴
∴
∴
x2 = 64 (simplify)
√x2 = √64 (take square root of both sides)
x = ±8 (simplify)
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
2) -x2 + 10 = 0
∴-x2 + 10 – 10 = 0 - 10 (undo addition by subtracting)
∴
∴
-x2 = -10
(simplify)
-x2/-1 = -10/-1 (undo multiplication by dividing)
∴
x2 = 10
∴
√x2 = √10
(simplify)
(take square root of both sides)
x = ±3.16 (simplify)
∴
3) 4x2 + 20 = 0
∴4x2 + 20 - 20 = 0 - 20 (undo addition by subtracting)
∴
∴
4x2 = -20
(simplify)
4x2/4 = -20/4 (undo multiplication by dividing)
∴
x2 = -5
∴
√x2 = √-5
(simplify)
(take square root of both sides)
∴There are no real solutions to the equation
Let’s work through some more examples of solving quadratic equations when b equals 0.
Example one requires us to solve x squared subtract 64 equals zero. The first step in solving this is to
undo the subtraction of 64 by adding 64 to both sides, which results in the simplified equation x
squared equals 64. Since a is equal to 1 in this equation there is no need to divide through by
anything, which means we simply need to take the square root of both sides of the equation to give a
result of either positive or negative 8.
Example two requires us to solve negative x squared plus 10 equals zero. The first step in solving this
is to undo the addition of 10 by subtracting 10 from both sides, which results in the simplified
equation negative x squared equals negative 10. We then need to divide both sides of the equation
through by negative 1, which results in the equation x squared equals 10. Finally, taking the square
root of both sides of the equation gives a result of either positive or negative 3.16.
Example three requires us to solve negative 4x squared plus 20 equals zero. The first step in solving
this is to undo the addition of 20 by subtracting 20 from both sides, which results in the simplified
equation 4x squared equals negative 20. We then need to divide both sides of the equation through
by 4, which results in the equation x squared equals negative 5. Note that we cannot solve the
equation in this instance since there are no real square roots of negative 5, so we simply state that
there are no real solutions to the equation.
Note that once you get the hang of rearranging and solving equations you may be able to leave out
some steps of working when writing up your solution, but for the time being it is best to include
everything so that any mistakes can be easily identified.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
Activity 1: Practice Questions
Click on the Activity 1 link in the right-hand part of this screen.
Now have a go at solving quadratic equations when b equals 0 on your own by working through some
practice questions.
Solving Quadratic Equations by Factorising
The second type of quadratic equation you should be able to solve is a quadratic equation that can be
factorised using one or two sets of brackets. You can solve equations like this by following these
steps:
1) Factorise the quadratic equation.
For example to solve the quadratic equation x2 + 7x + 12 = 0, factorise it as (x + 3)(x + 4) = 0
2) Solve the equation by solving the factors, since at least one of these must be equal to zero in
order for the equation to equal zero.
For example to solve the quadratic equation that can be factorised as (x + 3)(x + 4) = 0, note that
either
x + 3 = 0 or x + 4 = 0. Therefore x = -3 or x = -4
The second type of quadratic equation you should be able to solve is a quadratic equation that can be
factorised using one or two sets of brackets. You can solve equations like this by following two steps.
The first step is to factorise the quadratic equation. For example, to solve the quadratic equation x
squared plus 7x plus 12 equals 0, you should factorise it so that the equation becomes x plus 3, in
brackets, times x plus 4, also in brackets, equals 0. Note that if you need revision on how to factorise
quadratic equations, please refer to the Expanding Brackets and Factorising topic where this is covered.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
The second and final step is to then solve the equation by solving the factors, since at least one of
these must be equal to zero in order for the equation to equal zero. For example to solve the
quadratic equation that can be factorised as x plus 3, in brackets, times x plus 4, also in brackets,
equals 0, note that either x plus 3 must be equal to zero, or x plus 4 must be equal to zero. Therefore
solving both of these equations tells us that x must be equal to either negative 3 or negative 4. For
more information on how to solve linear equations such as this, please review the Rearranging and
Solving Linear Equations topic.
Examples: Solving Quadratic Equations by Factorising
1) x2 + 7x + 6 = 0
∴ (x + 6)(x + 1) = 0
∴ x + 6 = 0 or x + 1 = 0
∴ x = -6 or x = -1
2) x2 - 4x - 5 = 0
∴ (x + 1)(x - 5) = 0
∴ x + 1 = 0 or x - 5 = 0
∴ x = -1 or x = 5
3) x2 + 3x = 0
∴ x(x + 3) = 0
∴ x = 0 or x + 3 = 0
∴ x = 0 or x = -3
Let’s work through some more examples of solving quadratic equations by factorising.
The first example requires us to solve x squared plus 7x plus 6 equals 0, and the first step in doing this is
to factorise the equation using two sets of brackets, one of which contains x plus 6 and the other of which
contains x plus 1. It follows that either x plus 6 or x plus 1 must be equal to 0, in order for the product of
them to be equal to zero, and solving these equations gives x equal to either negative 6 or negative 1.
The second example requires us to solve x squared subtract 4x subtract 5 equals 0, and the first step in
doing this is to factorise the equation using two sets of brackets, one of which contains x plus 1 and the
other of which contains x subtract 5. It follows that either x plus 1 or x subtract 5 must be equal to 0, in
order for the product of them to be equal to zero, and solving these equations gives x equal to either
negative 1 or positive 5.
The third example requires us to solve x squared plus 3x equals 0, and the first step in doing this is to
factorise the equation so that it becomes x times, in brackets, x plus 3, equals zero. It follows that either
x or x plus 3 must be equal to 0, in order for the product of them to be equal to zero, and hence x is
either equal to 0 or to negative 3.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
Activity 2: Practice Questions
Click on the Activity 2 link in the right-hand part of this screen.
Now have a go at solving quadratic equations by factorising on your own by working through some
practice questions.
Solving Quadratic Equations using the Quadratic Formula
While factorising a quadratic equation can be a straightforward way of solving it, it is not always the
case that the equation can be factorised easily- or indeed at all.
In this case, you can use the quadratic formula to solve the equation. This states that for a quadratic
equation of the form y = ax2 + bx + c, the value of x is given by:
−𝑏𝑏 ± √𝑏𝑏2 − 4𝑎𝑎𝑎𝑎
2𝑎𝑎
2
For example, for the quadratic equation x + 2x – 24 = 0, the quadratic formula tells us that:
𝑥𝑥 =
𝑥𝑥 =
−2 ± �22 − 4(1)(−24)
2(1)
−2 ± √4 + 96
2
−2 ± √100
=
2
−2 ± 10
=
2
8
−12
= 𝑜𝑜𝑜𝑜
2
2
= 4 𝑜𝑜𝑜𝑜 − 6
=
While factorising a quadratic equation can be a straightforward way of solving it, it is not always the
case that the equation can be factorised easily, or indeed at all. In this case, you can use the
quadratic formula to solve the equation.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
The quadratic formula states that for a quadratic equation of the form y equals ax squared plus bx
plus c, the value of x is equal to negative b plus or minus the square root of b squared subtract 4ac,
all divided by 2a.
For example, for the quadratic equation x squared plus 2x subtract 24 equals zero, inserting 1 in place
of a, 2 in place of b and negative 24 in place of c in the quadratic formula gives x equal to negative 2
plus or minus the square root of 2 squared, minus 4 times 1 times negative 24, all divided by 2 times
1. Simplifying this equation by evaluating the square and multiplication gives x equal to negative 2
plus or minus the square root of 4 plus 96, all divided by 2, which simplifies to negative 2 plus or
minus the square root of 100 all divided by 2. Evaluating the square root then gives x equal to
negative 2 plus or minus 10 all divided by 2, which simplifies to either 8 over 2 or negative 12 over 2,
depending on whether the 10 is added to or subtracted from negative 2. Hence the solution to the
equation is either x equals 4 or x equals negative 6.
Examples: Solving Quadratic Equations using the Quadratic Formula
1) x2 + 7x + 10 = 0
𝑥𝑥 =
=
−7 ± �72 − 4(1)(10)
2(1)
−7 ± √49 − 40
2
−7 ± √9
2
−7 ± 3
=
2
−4
−10
=
𝑜𝑜𝑜𝑜
2
2
=
= −2 𝑜𝑜𝑜𝑜 − 5
2) x2 - 2x - 15 = 0
𝑥𝑥 =
=
2 ± �(−2)2 − 4(1)(−15)
2(1)
2 ± √4 + 60
2
2 ± √64
2
2±8
=
2
10
−6
=
𝑜𝑜𝑜𝑜
2
2
=
= 5 𝑜𝑜𝑜𝑜 − 3
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
Let’s work through some more examples of solving quadratic equations using the quadratic formula.
The first example requires you to solve the quadratic equation x squared plus 7x plus 10 equals 0, so
inserting 1 in place of a, 7 in place of b and 10 in place of c in the quadratic formula gives x equal to
negative 7 plus or minus the square root of 7 squared, minus 4 times 1 times 10, all divided by 2
times 1. Simplifying this equation by evaluating the square and multiplication gives x equal to
negative 7 plus or minus the square root of 49 subtract 40, all divided by 2, which simplifies to
negative 7 plus or minus the square root of 9 all divided by 2. Evaluating the square root then gives x
equal to negative 7 plus or minus 3 all divided by 2, which simplifies to either negative 4 over 2 or
negative 10 over 2, depending on whether the 3 is added to or subtracted from negative 7. Hence the
solution to the equation is either x equals negative 2 or x equals negative 5.
The second example requires you to solve the quadratic equation x squared subtract 2x subtract 15
equals 0, so inserting 1 in place of a, negative 2 in place of b and negative 15 in place of c in the
quadratic formula gives x equal to 2 plus or minus the square root of negative 2 squared, minus 4
times 1 times negative 15, all divided by 2 times 1. Simplifying this equation by evaluating the square
and multiplication gives x equal to 2 plus or minus the square root of 4 plus 60, all divided by 2, which
simplifies to 2 plus or minus the square root of 64 all divided by 2. Evaluating the square root then
gives x equal to 2 plus or minus 8 all divided by 2, which simplifies to either 10 over 2 or negative 6
over 2, depending on whether the 8 is added to or subtracted from 2. Hence the solution to the
equation is either x equals 5 or x equals negative 3.
Activity 3: Practice Questions
Click on the Activity 3 link in the right-hand part of this screen.
Now have a go at solving quadratic equations using the quadratic formula on your own by working
through some practice questions.
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Better Math – Numeracy Basics
Algebra - Rearranging and Solving Quadratic Equations
End of Topic
Congratulations, you have completed this topic.
You should now have a better understanding of rearranging and solving quadratic equations.
Congratulations, you have completed this topic.
You should now have a better understanding of rearranging and solving quadratic equations.
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