SME
Grade 9 Math · Problem type
Difference of two squares
Use this for “Factor 9x² − 25.”
Diagram
The tiles of 9x² − 25 make a rectangle (3x + 5) by (3x − 5): (3x + 5)(3x − 5) = 9x² − 25.
15 positive and negative x tiles cancel as zero pairs.
This is an example. Type your own values to start a new problem.
a
x²
−
c
=
(
p
x
+
q
)(
p
x
−
q
)
u
Number in the second bracket
worked out (not typed)
z
x terms, which cancel
worked out (not typed)
Formulas
p
=
√
a
3
=
√
9
q
=
√
c
5
=
√
25
u
=
−
q
−5 = −5
z
=
p
×
u
+
q
×
p
0 = 3 × (−5) + 5 × 3
Assumptions
•
p²x² − q² = (px + q)(px − q): the two middle terms cancel.
•
Both terms must be perfect squares, and they must be subtracted.
•
A sum of two squares, such as x² + 4, does not factor over the integers.
Step-by-step
WE KNOW
a = 9, c = 25
FIND
square root of a (p), square root of c (q), number in the second bracket (u), x terms, which cancel (z)
Step 1 · Find square root of a (p)
Use p = √a
The
first
term
is
a
square:
(px)
2
=
ax².
p
=
√
9
p
=
3
Step 2 · Find square root of c (q)
Use q = √c
The
number
taken
away
is
a
square
too:
q
2
=
c
.
q
=
√
25
q
=
5
Step 3 · Find number in the second bracket (u)
Use u = −q
One
bracket
adds
q
and
the
other
takes
it
away.
u
=
−5
Step 4 · Find x terms, which cancel (z)
Use z = p × u + q × p
Check the middle: the outer and inner x terms are opposites, so they cancel.
z
=
3
×
(−5)
+
5
×
3
z
=
−15
+
15
z
=
0
→
9x²
−
25
=
(3x
+
5)(3x
−
5)
Check
Put the numbers back in. Both sides should match.
3
=
√
9
✓
5
=
√
25
✓
5 + (−5) = 0 ✓
0 = 3 × (−5) + 5 × 3 ✓
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