SME
Grade 9 Math · Problem type
Solve by square roots
Use this for “(x − 3)² = 25” or “2(x − 1)² = 18” (a = 1 when there is no number in front).
Chart
Vertex (3, −25), axis of symmetry x = 3
Zeros: x = −2, x = 8
This is an example. Type your own values to start a new problem.
a
(x
+
(
p
)
)²
=
q
For (x − 3)², type −3 in the bracket: x − 3 is x + (−3).
Q
Right side after dividing by a
worked out (not typed)
x₁
Smaller root
worked out (not typed)
x₂
Larger root
worked out (not typed)
Formulas
Q
=
q
a
25
=
25
1
x₁
=
−
p
−
√
Q
−2
=
−(−3)
−
√
25
x₂
=
−
p
+
√
Q
8
=
−(−3)
+
√
25
Assumptions
•
Divide both sides by the number in front first: a(x + p)² = q gives (x + p)² = q ÷ a.
•
Take the square root of both sides: there are two roots, + and −.
•
A negative right side has no real solution.
Step-by-step
WE KNOW
a = 1, p = −3, q = 25
FIND
right side after dividing by a (Q), smaller root (x₁), larger root (x₂)
Step 1 · Find right side after dividing by a (Q)
Use Q = q ÷ a
Divide
both
sides
by
a
,
so
the
square
stands
alone:
(x +
p
)
2
=
Q
.
Q
=
25
1
Q
=
25
Step 2 · Find smaller root (x₁)
Use x₁ = −p − √Q
Take
the
square
root
of
both
sides,
x
+
p
=
±√
Q
,
then
take
p
away.
The
minus
root
first.
x₁
=
−(−3)
−
√
25
x₁
=
3
−
√
25
x₁
=
3
−
5
x₁
=
−2
Step 3 · Find larger root (x₂)
Use x₂ = −p + √Q
The plus root.
x₂
=
−(−3)
+
√
25
x₂
=
3
+
√
25
x₂
=
3
+
5
x₂
=
8
Check
Put the numbers back in. Both sides should match.
25
=
25
1
✓
−2
=
−(−3)
−
√
25
✓
8
=
−(−3)
+
√
25
✓
← Previous lesson
Solving quadratic equations: factoring, square roots, completing the square and the quadratic formula
Next lesson →
Complete the square
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.