For a certain positive number y, the product of y and the quantity 2y + 5 is equal to 12. … — ACT

For a certain positive number $y$ , the product of $y$ and the quantity $2y + 5$ is equal to $12$ . What is the value of $y$ ?

Answer: 1.5

The positive value of

y

1.5

The correct answer is

1.5

. By translating the word problem, we obtain

y(2y + 5) = 12

. Expanding this gives

2y^2 + 5y = 12

, and subtracting

12

from both sides yields the standard quadratic equation

2y^2 + 5y - 12 = 0

. Factoring this expression gives

(2y - 3)(y + 4) = 0

. Setting the first factor to zero yields

y = 1.5

, which is positive and therefore satisfies the given condition.

Write the equation representing the relationship.

y(2y + 5) = 12

To translate the verbal description into an algebraic equation.

Distribute

y

and set the equation equal to zero.

2y^2 + 5y - 12 = 0

To put the quadratic equation into standard form

ay^2 + by + c = 0

so it can be factored.

Factor the quadratic equation.

(2y - 3)(y + 4) = 0

To find the factors that multiply to give the quadratic expression.

Solve for

y

and apply the constraint.

y = 1.5

Setting the factors to zero gives

y = 1.5

and

y = -4

. Since the problem states

y

is positive, we select the positive root.

Solving a non-monic quadratic equation by factoring after translating a verbal description into algebra.

Question