Quadratic Word Problems: Projectiles, Area, and More

Quadratics show up whenever something is squared: the area of a shape, the height of a thrown ball, or revenue when price and sales both change. The algebra is the same every time; the skill is setting up the equation and deciding which solution makes sense.

L'idée en bref

Translate the story into an equation, rearrange it to equal zero, solve by factoring or the quadratic formula, then reject any answer that is impossible in the real situation.

Les étapes

1. Define a variable and write what it stands for, with units.
2. Build the equation from the story, then rearrange it to standard form = 0.
3. Solve by factoring, completing the square, or the quadratic formula.
4. Check each solution in the story and keep only the ones that make sense.

Exemple résolu

A ball is thrown from a 48-foot platform. Its height after t seconds is h = −16t² + 32t + 48. When does it hit the ground?

Solution complète

Explication étape par étape

Étape 1

The ball hits the ground when its height is 0, so set the expression equal to 0.

Étape 2

Divide every term by −16 to make the numbers smaller.

Étape 3

Factor: two numbers that multiply to −3 and add to −2 are −3 and 1.

Étape 4

t = 3 or t = −1. Time cannot be negative, so the ball lands after 3 seconds.

Erreur fréquente

Do not report both solutions automatically. Negative times, negative lengths, and fractional numbers of people usually have to be rejected.

Exercices d'entraînement

1. A rectangle's length is 3 cm more than its width, and its area is 40 cm². Find its dimensions.
2. The product of two consecutive positive integers is 72. Find them.
3. A ball's height is h = −16t² + 64t. What is its maximum height?

Corrigés

1. 5 cm by 8 cm
2. 8 and 9
3. 64 feet, at t = 2 seconds

Demandez à PennPaper de vous l'expliquer en direct

Si les étapes vous semblent claires mais que vous bloquez encore, lancez une séance PennPaper et demandez au tuteur de dessiner le problème sur le tableau blanc. Voir les symboles bouger un à un, c'est souvent ce qui fait tilt.

Guides associés

• La formule des racines (discriminant) expliquée
• Comment factoriser un trinôme du second degré
• Vertex Form and How to Graph a Parabola
• Comment mettre un problème en équation