year 8 maths

2F
Solving simple quadratic equations
87
2F Solving simple quadratic equations
Most of the equations you have worked with so far are called linear equations like 2x − 3 = 7 and
5(a + 2) = 7(a − 1), where the power of the pronumeral is 1 and there is usually a single solution.
Another type of equation is of the form x 2 = c and this is an example of a simple quadratic equation.
Note that the power of the pronumeral x is 2 . Depending on the value of c, x can have zero, one or two
solutions. These types of equations appear frequently in mathematics and in problems involving distance,
area, graphs and motion.
Let’s start: How many solutions?
Consider the equation x 2 = c . How many values of x can you think of that satisfy the equation when:
• c = 0 ?
• c = 9 ?
• c = −4 ?
What conclusions can you come to regarding the number of solutions for x depending on the value
of c ?
■ Simple quadratic equations of the form x 2 = c:
• x 2 = 9 has two solutions because 9 is a positive number.
x 2 = 9
x = √9,
x = 3,
x = −√9
x = −3
Note: 3 2 = 9 and (−3) 2 = 9 .
x = ±3, where ±3 represent both solutions.
• x 2 = 0 has one solution (x = 0) because 0 2 = 0 .
• x 2 = −9 has no solution because the square of any number is 0 or positive.
Key ideas
Example 9 Solving x 2 = c when c > 0
Solve the following equations. Round to two decimal places in part b by using a calculator to assist.
a x 2 = 81 b x 2 = 23
SOLUTION
EXPLANATION
a x = 9 or x = −9 The equation has two solutions because 81 is a
positive number. 9 2 = 81 and (−9) 2 = 81.
b
x = √23 = 4.80 (to 2 decimal places) or
x = −√23 = −4.80 (to 2 decimal places)
The number 23 is not a perfect square so √23 can
be rounded if required.
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
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