Applications turn real situations, like a thrown ball or a measured area, into quadratic equations.
A ball thrown at an angle traces a parabola, and its height reaching zero tells you when it lands.
You pick an unknown, form the equation, and solve, dropping any root that breaks a condition like a negative length.
Change the speed and angle to explore the path, then stretch a rectangle to build its area equation.
Projectile Motion — Quadratics in Nature
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Projectile path and the quadratic equation that describes it
20 m/s
45°
🏀 Projectile Motion & Quadratics
①Throwing a ball traces a parabola → y = ax²+bx+c
②When does it hit the ground? → y=0 → quadratic equation
③Maximum height → y-coordinate of the vertex
Area Problems — Modeling Geometry
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Express how a rectangle's area grows as a function
3
📐 Strategy for Area Problems
①Define unknown x (the increase, decrease, etc.)
②Express area as a function of x
③Set up an equation matching the condition
④Solve and check (must be positive, etc.)
Number Problems
Consecutive Integers
n(n+1) = 90 → n²+n−90 = 0
Consecutive integers: n and n+1
🔢 Common Number Problems
①Product of two consecutive naturals = A → n(n+1) = A
②Square of a number minus 3 = … → x²−3 = …
③Two-digit number digits product → (10a+b), ab = …
4 Steps to Solve Word Problems
Procedure
Set unknown → Build equation → Solve → Verify
Always follow this order
⚠️ Why Verification Matters
①Quadratics may yield two roots
②Drop roots that don't fit the conditions
③'Find the natural number' → drop negatives
④'Find the length' → drop negatives
⑤Always plug back into the original problem
Work It Out
Example 1
The product of two consecutive natural numbers is 56. Find them.
1
Let the smaller be n; then n(n+1)=56.
n(n+1)=56 ⇒ n²+n−56=0
2
Factor: (n−7)(n+8)=0. Since n is natural, n=7.
(n−7)(n+8)=0 ⇒ n=7
▸ 7 and 8
The natural-number condition rules out n=−8.
Example 2
For a positive number x, x(x+3)=40. Find x.
1
Expand and move everything to one side.
x²+3x−40=0
2
Factor: (x−5)(x+8)=0. Since x>0, x=5.
(x−5)(x+8)=0 ⇒ x=5
▸ x = 5
The positive condition rules out x=−8.
Exam Wrap-up
Application Templates
Consec int: n, n+1 / Area: w×h / Time: h=v₀t−5t²
Memorize how to set the unknown by problem type
G9 school-exam type
An object is launched from the ground; its height after t seconds is h=−5t²+20t (m). After how many seconds does it return to the ground?
①1 s
②2 s
③4 s
④5 s
⑤8 s
▸ ③ 4 s
1
On the ground means h=0.
−5t²+20t=0
2
From −5t(t−4)=0 with t>0, t=4.
−5t(t−4)=0 ⇒ t=4
🎯 Exam Points
①Set the unknown as what you want to find
②Move everything to one side (= 0)
③Check roots against conditions (natural, positive)