SME
Grade 9 Math · Problem type
Quadratic inequality
Use this for “Solve x² − 2x − 8 < 0” or “x² − x − 6 ≥ 0.”
Chart
f(x) < 0 where the curve is below the x-axis: −2 < x < 4.
Open circles: the zeros make it 0, left out
Discriminant: 36
Test number: 0
Is the test true?: true
This is an example. Type your own values to start a new problem.
x²
+
(
b
)
x
+
(
c
)
0
D
Discriminant
worked out (not typed)
x₁
Smaller zero
worked out (not typed)
x₂
Larger zero
worked out (not typed)
x₀
Test number
example
h
Is the test true?
worked out (not typed)
Formulas
D
=
b
2
−
4
×
c
36
=
(−2)
2
−
4
×
(−8)
x₁
=
−
b
− √
D
2
−2
=
−(−2) −
√
36
2
x₂
=
−
b
+ √
D
2
4
=
−(−2) +
√
36
2
test
x₀
in
x
2
+
bx
+
c
(sign
s
)
0:
h
test
0
in
x
2
+
(−2)x
+
(−8)
(sign
<)
0:
true
Assumptions
•
Solve the equation x² + bx + c = 0 first: its zeros split the number line.
•
The parabola opens up: below the x-axis between its zeros, above it outside them.
•
The zeros make it 0: ≤ and ≥ take them in, < and > leave them out.
Step-by-step
WE KNOW
b = −2, c = −8, s: <, x₀ = 0
FIND
discriminant (D), smaller zero (x₁), larger zero (x₂), is the test true? (h)
Step 1 · Find discriminant (D)
Use D = b² − 4 × c
The discriminant, with a = 1.
D
=
(−2)
2
−
4
×
(−8)
D
=
4
−
4
×
(−8)
D
=
4
−
(−32)
D
=
36
Step 2 · Find smaller zero (x₁)
Use x₁ = (−b − √D) ÷ 2
Solve
x
2
+
bx
+
c
=
0
first:
the
quadratic
formula
with
a
=
1
and
the
minus
sign.
x₁
=
−(−2) −
√
36
2
x₁
=
2 −
√
36
2
x₁
=
2 − 6
2
x₁
=
−
4
2
x₁
=
−2
Step 3 · Find larger zero (x₂)
Use x₂ = (−b + √D) ÷ 2
The plus sign gives the larger zero.
x₂
=
−(−2) +
√
36
2
x₂
=
2 +
√
36
2
x₂
=
2 + 6
2
x₂
=
8
2
x₂
=
4
→
−2
<
x
<
4
Step 4 · Find is the test true? (h)
Use test x₀ in x² + bx + c (sign s) 0: h
Put the test number into the left side: is it true? Then it is a solution.
(0)
2
−
2(0)
−
8
=
−8
−8 < 0 is true
h
=
true
Check
Put the numbers back in. Both sides should match.
36
=
(−2)
2
−
4
×
(−8)
✓
−2
=
−(−2) −
√
36
2
✓
4
=
−(−2) +
√
36
2
✓
−8 < 0, so the test number is a solution ✓
← Previous lesson
Complete the square
Next lesson →
One-variable statistics: histograms, box plots, standard deviation and outliers
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Solving quadratic equations: factoring, square roots, completing the square and the quadratic formula
Main lesson
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).
Complete the square
Use this for “Solve x² + 6x − 7 = 0 by completing the square.”