SME
Grade 9 Math · Problem type
A line and a parabola
Use this for “Solve y = x² − 2x − 3 and y = x + 1.”
Chart
Set them equal: x² − 3x − 4 = 0
They cross twice, at (−1, 0) and (4, 5)
This is an example. Type your own values to start a new problem.
y
=
a
x²
+
(
p
)
x
+
(
q
)
y
=
m
x
+
k
(
x₁
,
y₁
)
and
(
x₂
,
y₂
)
B
x coefficient after gathering, p − m
worked out (not typed)
C
Number after gathering, q − k
worked out (not typed)
Formulas
B
=
p
−
m
−3 = −2 − 1
C
=
q
−
k
−4 = −3 − 1
x₁
=
−
B
−
√
B
2
− 4 ×
a
×
C
2 ×
a
−1
=
(−(−3)
−
√(
(−3)
2
−
4
×
1
×
(−4)))
÷
(2
×
1)
y₁
=
m
×
x₁
+
k
0 = 1 × (−1) + 1
x₂
=
−
B
+
√
B
2
− 4 ×
a
×
C
2 ×
a
4
=
(−(−3)
+
√(
(−3)
2
−
4
×
1
×
(−4)))
÷
(2
×
1)
y₂
=
m
×
x₂
+
k
5 = 1 × 4 + 1
Assumptions
•
At a crossing both equations give the same y, so set ax² + px + q equal to mx + k.
•
Gather everything on one side: ax² + (p − m)x + (q − k) = 0, then use the quadratic formula.
•
Two roots: the line cuts the parabola twice; one: it touches; none: it misses.
Step-by-step
WE KNOW
a = 1, p = −2, q = −3, m = 1, k = 1
FIND
x coefficient after gathering, p − m (B), number after gathering, q − k (C), left crossing x (x₁), left crossing y (y₁), right crossing x (x₂), right crossing y (y₂)
Step 1 · Find x coefficient after gathering, p − m (B)
Use B = p − m
Set the right sides equal and take mx from both sides: the x terms combine.
B
=
−2
−
1
B
=
−3
Step 2 · Find number after gathering, q − k (C)
Use C = q − k
Take
k
from
both
sides
too:
everything
is
on
one
side,
=
0.
C
=
−3
−
1
C
=
−4
→
x
2
−
3x
−
4
=
0
Step 3 · Find left crossing x (x₁)
Use x₁ = (−B − √(B² − 4 × a × C)) ÷ (2 × a)
The quadratic formula with the minus sign: the smaller root.
x₁
=
(−(−3)
−
√(
(−3)
2
−
4
×
1
×
(−4)))
÷
(2
×
1)
x₁
=
(−(−3)
−
√(9
−
4
×
(−4)))
÷
2
x₁
=
(−(−3)
−
√(9
−
(−16)))
÷
2
x₁
=
−(−3) −
√
25
2
x₁
=
3 −
√
25
2
x₁
=
3 − 5
2
x₁
=
−
2
2
x₁
=
−1
Step 4 · Find left crossing y (y₁)
Use y₁ = m × x₁ + k
Put
x₁
into
the
line
(the
simpler
equation)
for
its
y.
y₁
=
1
×
(−1)
+
1
y₁
=
0
Step 5 · Find right crossing x (x₂)
Use x₂ = (−B + √(B² − 4 × a × C)) ÷ (2 × a)
The plus sign: the second crossing (the same point when the line only touches).
x₂
=
(−(−3)
+
√(
(−3)
2
−
4
×
1
×
(−4)))
÷
(2
×
1)
x₂
=
(−(−3)
+
√(9
−
4
×
(−4)))
÷
2
x₂
=
(−(−3)
+
√(9
−
(−16)))
÷
2
x₂
=
−(−3) +
√
25
2
x₂
=
3 +
√
25
2
x₂
=
3 + 5
2
x₂
=
8
2
x₂
=
4
Step 6 · Find right crossing y (y₂)
Use y₂ = m × x₂ + k
Put
x₂
into
the
line
for
its
y.
y₂
=
1
×
4
+
1
y₂
=
5
Check
Put the numbers back in. Both sides should match.
−3 = −2 − 1 ✓
−4 = −3 − 1 ✓
−1
=
(−(−3)
−
√(
(−3)
2
−
4
×
1
×
(−4)))
÷
(2
×
1)
✓
0 = 1 × (−1) + 1 ✓
4
=
(−(−3)
+
√(
(−3)
2
−
4
×
1
×
(−4)))
÷
(2
×
1)
✓
5 = 1 × 4 + 1 ✓
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