SME
Grade 11 Math · Problem type
Equations in quadratic form
Use this for “Solve x⁴ − 13x² + 36 = 0” when every root is real.
Chart
Zeros: x = −3, x = −2, x = 2, x = 3
This is an example. Type your own values to start a new problem.
x⁴
+
(
b
)
x²
+
c
=
0
u²
+
(
b
)
u
+
c
=
0
Put u = x²: the second line is the same equation, a quadratic in u.
u₁
Larger u
worked out (not typed)
u₂
Smaller u
worked out (not typed)
x₁
Positive root from u₁
worked out (not typed)
x₂
Positive root from u₂
worked out (not typed)
Formulas
u₁
=
−
b
+
√
b
2
− 4 ×
c
2
9
=
(−(−13)
+
√(
(−13)
2
−
4
×
36))
÷
2
u₂
=
−
b
−
√
b
2
− 4 ×
c
2
4
=
(−(−13)
−
√(
(−13)
2
−
4
×
36))
÷
2
x₁
=
√
u₁
3
=
√
9
x₂
=
√
u₂
2
=
√
4
Assumptions
•
Let u = x²: then x⁴ + bx² + c = 0 is the quadratic u² + bu + c = 0.
•
Solve for u, then x = ±√u for each u.
•
A negative u gives no real x, so this page needs both u to be 0 or more.
Step-by-step
WE KNOW
b = −13, c = 36
FIND
larger u (u₁), smaller u (u₂), positive root from u₁ (x₁), positive root from u₂ (x₂)
Step 1 · Find larger u (u₁)
Use u₁ = (−b + √(b² − 4 × c)) ÷ 2
The
quadratic
formula
for
u
2
+
bu
+
c
=
0,
with
the
plus
sign.
u₁
=
(−(−13)
+
√(
(−13)
2
−
4
×
36))
÷
2
u₁
=
13 +
√
25
2
u₁
=
13 + 5
2
u₁
=
9
Step 2 · Find smaller u (u₂)
Use u₂ = (−b − √(b² − 4 × c)) ÷ 2
The same formula with the minus sign.
u₂
=
(−(−13)
−
√(
(−13)
2
−
4
×
36))
÷
2
u₂
=
13 −
√
25
2
u₂
=
13 − 5
2
u₂
=
4
Step 3 · Find positive root from u₁ (x₁)
Use x₁ = √u₁
x
2
=
u₁
,
so
x
=
±√
u₁
:
this
is
the
positive
one.
x₁
=
√
9
x₁
=
3
Step 4 · Find positive root from u₂ (x₂)
Use x₂ = √u₂
x
2
=
u₂
,
so
x
=
±√
u₂
.
x₂
=
√
4
x = ±3 or x = ±2
x₂
=
2
Check
Put the numbers back in. Both sides should match.
9
=
(−(−13)
+
√(
(−13)
2
−
4
×
36))
÷
2
✓
4
=
(−(−13)
−
√(
(−13)
2
−
4
×
36))
÷
2
✓
3
=
√
9
✓
2
=
√
4
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Factoring polynomials: GCF, trinomials and special products
Solving quadratic equations: factoring, square roots, completing the square and the quadratic formula