Mathovia

Math Practice

Quadratic Application Practice Problems

Twelve hand-built quadratic word problems with full worked solutions. Each shows the variable choice, the quadratic, the solving step, and — the part that matters — which of the two roots gets rejected as physically impossible.

  1. 1.

    A rectangle is 4 cm longer than it is wide, with an area of 96 cm². Find its dimensions.

    Show full solution

    $$ w(w + 4) = 96 \quad\Longrightarrow\quad w^{2} + 4w - 96 = 0 \quad\Longrightarrow\quad (w + 12)(w - 8) = 0 $$

    Reject \(w = -12\). Width \(8\) cm, length \(12\) cm.

  2. 2.

    The product of two consecutive integers is 156. Find them.

    Show full solution

    $$ x(x + 1) = 156 \quad\Longrightarrow\quad x^{2} + x - 156 = 0 \quad\Longrightarrow\quad (x + 13)(x - 12) = 0 $$

    \(x = 12\) gives \(12, 13\); \(x = -13\) gives \(-13, -12\).

  3. 3.

    The sum of a number and its square is 90. Find the number.

    Show full solution

    $$ x + x^{2} = 90 \quad\Longrightarrow\quad x^{2} + x - 90 = 0 \quad\Longrightarrow\quad (x + 10)(x - 9) = 0 $$

    \(x = 9\) or \(x = -10\); both check.

  4. 4.

    A number is 6 less than its square. Find the number.

    Show full solution

    $$ x = x^{2} - 6 \quad\Longrightarrow\quad x^{2} - x - 6 = 0 \quad\Longrightarrow\quad (x - 3)(x + 2) = 0 $$

    \(x = 3\) or \(x = -2\); both check.

  5. 5.

    A ball is thrown up from ground level at 64 ft/s. Using \(h = -16t^{2} + 64t\), when is it 48 ft high?

    Show full solution

    $$ -16t^{2} + 64t = 48 \quad\Longrightarrow\quad t^{2} - 4t + 3 = 0 \quad\Longrightarrow\quad (t - 1)(t - 3) = 0 $$

    \(t = 1\) s (rising) and \(t = 3\) s (falling). Both valid.

  6. 6.

    A rectangular garden is 3 times as long as it is wide, with an area of 300 m². Find the dimensions.

    Show full solution

    $$ w(3w) = 300 \quad\Longrightarrow\quad 3w^{2} = 300 \quad\Longrightarrow\quad w^{2} = 100 \quad\Longrightarrow\quad w = 10 $$

    Width \(10\) m, length \(30\) m.

  7. 7.

    The product of two consecutive odd integers is 195. Find them.

    Show full solution

    $$ x(x + 2) = 195 \quad\Longrightarrow\quad x^{2} + 2x - 195 = 0 \quad\Longrightarrow\quad (x + 15)(x - 13) = 0 $$

    \(13, 15\) or \(-15, -13\).

  8. 8.

    A triangle's base is 3 cm more than its height. Its area is 27 cm². Find the base and height.

    Show full solution

    $$ \frac{1}{2}h(h + 3) = 27 \quad\Longrightarrow\quad h^{2} + 3h - 54 = 0 \quad\Longrightarrow\quad (h + 9)(h - 6) = 0 $$

    Reject \(h = -9\). Height \(6\) cm, base \(9\) cm.

  9. 9.

    A photo 6 in by 8 in gets a uniform frame. The framed area is 120 sq in. Find the frame width.

    Show full solution

    $$ (6 + 2x)(8 + 2x) = 120 \quad\Longrightarrow\quad 4x^{2} + 28x + 48 = 120 \quad\Longrightarrow\quad x^{2} + 7x - 18 = 0 $$

    $$ (x + 9)(x - 2) = 0 \quad\Longrightarrow\quad x = 2 \text{ in} $$

  10. 10.

    A ball is thrown up from a height of 5 ft at 30 ft/s. When does it hit the ground? Use \(h = -16t^{2} + 30t + 5\).

    Show full solution

    $$ -16t^{2} + 30t + 5 = 0 \quad\Longrightarrow\quad 16t^{2} - 30t - 5 = 0 $$

    \(D = 900 + 320 = 1220\); \(t = \dfrac{30 \pm \sqrt{1220}}{32} = \dfrac{15 \pm \sqrt{305}}{16}\). \(\sqrt{305} \approx 17.46\), so \(t \approx 2.03\) s (reject the negative).

  11. 11.

    A path of uniform width surrounds a 20 ft by 30 ft pool. Pool plus path covers 936 sq ft. Find the path width.

    Show full solution

    $$ (20 + 2x)(30 + 2x) = 936 \quad\Longrightarrow\quad 4x^{2} + 100x + 600 = 936 \quad\Longrightarrow\quad x^{2} + 25x - 84 = 0 $$

    $$ (x + 28)(x - 3) = 0 \quad\Longrightarrow\quad x = 3 \text{ ft} $$

  12. 12.

    At price \(p\) dollars, a vendor sells \(80 - 4p\) items. What price gives \(\$400\) in revenue?

    Show full solution

    $$ p(80 - 4p) = 400 \quad\Longrightarrow\quad -4p^{2} + 80p - 400 = 0 \quad\Longrightarrow\quad p^{2} - 20p + 100 = 0 $$

    $$ (p - 10)^{2} = 0 \quad\Longrightarrow\quad p = \$10 $$
M
Written by
Mathovia Team
Editorial Team

Quadratic Application Practice Problems

The Applications of Quadratic Equations lesson works through area, projectile, integer-product, and revenue problems; this set is 12 to practice on, each fully solved.

How to use this set

These problems are hand-written, not auto-generated. Use the five-step method: identify the question, choose a variable, write the quadratic, solve it, and check both roots against the words — rejecting any impossible one. Then compare with the solution.

Common mistakes to watch for

Keeping a physically impossible root — a negative length, negative time, or fractional count.

Discarding a root too fast — a plain “consecutive integers” problem can have a negative-root pair.

Solving for and stopping without translating back to the dimensions.

Forgetting that a border adds , not , to a full dimension.

Where to go next

The finish is factoring or the Quadratic Formula Practice Problems. The linear versions of these (perimeter, distance, interest) are in the Word Problems Practice Problems. For the five-step method applied to quadratics, see the Applications of Quadratic Equations lesson.

Frequently Asked Questions

How do I know a word problem will lead to a quadratic?+

Look for a product of two quantities that both depend on the unknown: length times width, consecutive integers multiplied, price times quantity. Multiplying two expressions in produces an term.

Both roots are real. Which is the answer?+

Check each against the situation. Reject a negative length, a negative time, or a non-whole count. Sometimes both survive; often only one does.

What formula do projectile problems use?+

in feet and seconds, where is the initial upward velocity and the initial height. Setting to a value gives a quadratic in .

Why is this a curated set?+

Word problems don't reduce to a single auto-checkable number, and several of these have irrational answers. Each problem has a full solution instead.

For an area problem, do I solve for x or the dimensions?+

Solve the quadratic for , then translate back to the dimensions the problem asked for, discarding any solution that makes a dimension zero or negative.

Why does a border problem give a quadratic?+

The border adds width to all four sides, so both new dimensions are expressions in , and their product contains .

More practice problems

Quadratic Formula Practice Problems Quadratic Equations Practice Problems Algebra Word Problems Practice Problems