Quadratic Word Problems

Forming a Quadratic

Meaning: Words are translated into an equation with $x^{2}$.

Definition: A quadratic equation has highest power 2.

If length is $x + 3$ and breadth is x, area is x($x + 3$).

• Define x.
• Write related quantities.
• Use the condition.

Area Problems

Meaning: Area problems use multiplication of dimensions.

Definition: Rectangle are$a = l$ength x breadth.

If area is 70 and sides are x and $x + 3$, write $x(x + 3) = 70$.

• Lengths must be positive.

Consecutive Numbers

Meaning: Consecutive numbers follow one after another.

Definition: If one integer is x, the next is $x + 1$.

The product of consecutive integers is x($x + 1$).

• Consecutive even or odd numbers use x and $x + 2$.

Valid Roots

Meaning: Only answers that fit the situation are accepted.

Definition: A valid root satisfies both the equation and the context.

A negative length is rejected even if it solves the equation.

• Check both roots.
• Reject impossible values.
• Write the answer in words.

Easy

Question: The product of a number and the next number is 30. Find the positive number.

1. Let the number be x.
2. Next number is $x + 1$.
3. $x(x + 1) = 30$.
4. $x^{2} + x - 30 = 0$.
5. ($x + 6$)$(x - 5) = 0$.
6. Positive number is 5.

Medium

Question: A rectangle has breadth x cm and length $x + 4$ cm. Its area is 45 c$m^{2}$. Find x.

1. $x(x + 4) = 45$.
2. $x^{2} + 4$$x - 45 = 0$.
3. ($x + 9$)$(x - 5) = 0$.
4. $x = 5$ o$r -9$.
5. Breadth is positive, so $x = 5$.

Exam-level

Question: Two consecutive even numbers have product 168. Find the positive numbers.

1. Let the smaller even number be x.
2. Next even number is $x + 2$.
3. $x(x + 2) = 168$.
4. $x^{2} + 2$$x - 168 = 0$.
5. ($x + 14$)$(x - 12) = 0$.
6. The numbers are 12 and 14.