"Throw a ball straight up — how many seconds until it hits the ground?" That question is a quadratic equation. So is the flight time of a rocket, the sagging curve of a suspension bridge cable, and the moment a bullet lands in a game's physics engine. Wherever something follows a parabola, a quadratic equation holds the answer.
Flight time of a rocket or shell (h = v₀t − 4.9t²)
The shape of a bridge cable (a parabola describes the sag)
Game physics engines (working out where a projectile lands)
Maximizing profit (finding the peak of a revenue curve)
Key points
The Standard Form
A quadratic equation looks like ax² + bx + c = 0. The highest power of x is 2, and that is why it can have as many as two solutions.
Solving by Factoring
Factor the left side into (x − p)(x − q) = 0. A product can only be zero if one of its parts is zero, so x = p or x = q.
Solving by Completing the Square
Rewrite x² + bx as (x + b/2)² − (b/2)². Once the equation contains a single squared bracket, you can just take square roots.
The Quadratic Formula
This formula works on every quadratic equation, even the ones that refuse to factor. The part under the root, D = b² − 4ac, is called the discriminant, and its sign tells you how many real solutions to expect.
How Long Does the Rocket Stay Up?
Launch a rocket straight up at 20 m/s and its height is h = 20t − 5t². Setting h = 0 asks when it is at ground level, and the answers are t = 0 (the launch) and t = 4 seconds (the landing).
Check yourself 10questions
Tap an option to check
Q1 Solve x² − 5x + 6 = 0.
Factor it as (x−2)(x−3) = 0, so x = 2 or x = 3.
Q2 Solve x² − 9 = 0.
From x² = 9 you get x = ±√9 = ±3. Don't lose the negative root — (−3)² is also 9.
Q3 Solve x² + 4x + 4 = 0.
The left side is a perfect square: (x+2)² = 0. Both roots land on the same value, x = −2, which is called a double root.
Q4 Use the quadratic formula to solve 2x² + 3x − 2 = 0.
The discriminant is D = 9 + 16 = 25, and √25 = 5. So x = (−3 ± 5)/4, giving x = 1/2 or x = −2.
Q5 Solve x² − 4x = 0.
There is no constant term, so pull out the common factor: x(x−4) = 0. That gives x = 0 or x = 4.
Q6 What happens when the discriminant D = b² − 4ac is negative?
When D < 0 the formula asks for the square root of a negative number, so there is no real solution. In later grades these roots are handled as imaginary numbers.
Q7 A rectangle has a height plus width of 20 cm and an area of 96 cm². How tall is it?
Call the height x, so the width is 20 − x. Then x(20 − x) = 96, which becomes x² − 20x + 96 = 0 and factors as (x−8)(x−12) = 0. The two answers, 8 and 12, are just the height and width swapped.
Q8 A ball's height is h = 30t − 5t² (h in meters, t in seconds). When does it reach its highest point?
A parabola peaks on its axis of symmetry, which sits halfway between the two times the ball is at ground level (t = 0 and t = 6), so t = 3. Using calculus gives the same answer: the speed dh/dt = 30 − 10t is zero at t = 3.
Q9 For which value of k does x² − 6x + k = 0 have a double root?
A double root means the discriminant is exactly zero: D = 36 − 4k = 0, so k = 9. Check it — x² − 6x + 9 is (x−3)².
Q10 Solve 3(x − 1)² = 27.
Divide both sides by 3 to get (x−1)² = 9, then take square roots: x − 1 = ±3. So x = 4 or x = −2.
Jobs that use this
Aerospace Engineer$140k
Game Engineer$110k
Civil / Bridge Engineer$85k
Aerospace Engineer$140k
Works out the paths and flight times of rockets and aircraft using the equations of motion, which are quadratics.
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