grade 9 Applications of Quadratic Equations

Applications of Quadratic Equations

Applications turn a thrown ball or a measured plot into a quadratic equation. A ball thrown at an angle traces a parabola, and setting height to zero gives the landing time. 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 see the path, then stretch a rectangle to build its area equation.

Projectile Motion — Quadratics in Nature

Initial Speed v₀20 m/s

Launch Angle45°

🏀 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. Canvas uses g=10, default speed 20, angle 45°. The wrap-up h=v₀t−5t² is a vertical throw; the picture is angled. The rectangle starts 10×8 with default x=3

Area Problems — Modeling Geometry

x (extension, same units)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.

Let the smaller be n; then n(n+1)=56.

n(n+1)=56 ⇒ n²+n−56=0

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.

Expand and move everything to one side.

x²+3x−40=0

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

On the ground means h=0.

−5t²+20t=0

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)

④Match units in the final answer

⑤Watch out for '>', '≤' etc.

← Previous

Solving Quadratic Equations

Graphs of Quadratic Functions

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