Applications of Quadratic Equations

A word problem lands in quadratic territory whenever it multiplies two quantities that both depend on the unknown. Length times width, one integer times the next, price times quantity sold — each of those products contains an term once you substitute expressions in .

The setup is still the Word Problems five-step method. What changes is the finish: the equation is a quadratic, solved by factoring or the quadratic formula, and step five — checking the answer against the words — now often means rejecting one of the two solutions because it is physically impossible.

The Quadratic Application Formula

The last arrow is the one that separates quadratic applications from linear ones: you get two candidate answers and must decide which make sense.

Area and Geometry Problems

If a rectangle’s length is described in terms of its width, substituting into gives a quadratic.

The width is rejected; is the answer.

Projectile and Free-Fall Motion

is the initial upward velocity, the initial height. Setting to (hits the ground) or another value gives a quadratic in ; a negative time is rejected.

Consecutive Integer Products

Expanding gives , with roots and — both are valid integer pairs ( and ) unless the problem restricts to positive integers.

Revenue Problems

Revenue is price times quantity, and a demand relationship makes quantity depend on price:

Setting to a target value gives a quadratic in the price .

Worked Example A: A Rectangle

A rectangle is meters longer than it is wide, and its area is square meters. Find its dimensions.

Step 1–3 — let be the width; the length is :

Step 4 — factor:

Step 5 — reject (a width cannot be negative). The width is m and the length is m. Check: . ✓

Worked Example B: A Projectile

A ball is thrown upward from a height of feet with an initial velocity of ft/s. When does it hit the ground?

Step 1–3 — set in :

Step 4 — quadratic formula: , , ; discriminant .

Step 5 — , so s or s. Reject the negative time.

Answer: the ball hits the ground about seconds after release.

Worked Example C: Consecutive Integers

The product of two consecutive positive odd integers is . Find them.

Step 1–3 — let the first be ; the next odd integer is :

Step 4 — factor:

Step 5 — the problem says positive, so reject . The integers are and . Check: . ✓

Worked Example D: A Border

A rectangular garden is ft by ft. A gravel path of uniform width is added around the outside, and the total area (garden plus path) is sq ft. How wide is the path?

Step 1–3 — let be the path width. The outer rectangle is by :

Step 4 — expand and simplify:

Quadratic formula: , , ; discriminant .

Step 5 — reject the negative width. The path is ft wide. Check: . ✓

Common Mistakes to Avoid

• Keeping a physically impossible root. A negative length, negative time, or non-integer count is not an answer — discard it.
• Discarding a root too fast. In a pure “consecutive integers” problem with no positivity restriction, a negative root can give a legitimate integer pair.
• Solving for and stopping. If is the width, the problem still wants the length too.
• Forgetting that a border adds , not , to a full dimension. The path is on both sides.
• Sign error in . The (or ) is fixed and negative; is added, not subtracted.
• Not clearing to standard form before factoring or applying the formula.

Where This Shows Up Later

• Polynomial Inequalities. “For what widths is the area at least 50?” turns an application into an inequality.
• Graphing parabolas. A revenue function is a downward parabola; its vertex is the price that maximizes revenue.
• Optimization in calculus. The maximum-area and maximum-revenue questions previewed here are solved exactly with derivatives later.
• Systems with a quadratic. Some geometry problems combine a quadratic area equation with a linear perimeter equation.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. A rectangle’s length is cm more than its width. Its area is cm². Find the dimensions.

Problem 2. The product of two consecutive integers is . Find them.

Problem 3. A ball is dropped () from a height of ft. When does it land? Use .

Problem 4. The sum of a number and its square is . Find the number.

Problem 5. A rectangular field is twice as long as it is wide, with an area of m². Find the dimensions.

Problem 6. A rocket is launched from the ground at ft/s. Using , when is it ft high?

Problem 7. The product of two consecutive even integers is . Find them.

Problem 8. A photo in by in has a frame of uniform width. The framed area is sq in. Find the frame width.

Problem 9. A number is less than its square. Find the number.

Problem 10. A triangle’s height is cm more than its base. Its area is cm². Find the base and height. Use .

Problem 11. A store finds that at price dollars it sells items. What price gives a revenue of ?

Problem 12. A boat travels miles downstream and back in hours. Its speed in still water is mph. Find the current speed .

Quick Reference

The setup is the Word Problems method; the finish is factoring or the quadratic formula. Linear versions of these problems (perimeter, distance, interest) are in Applications of Linear Equations. More Algebra lessons are available too.