For the quadratic equation 0.4x^2 + bx - 4.8 = 0, where b is a constant, the sum of the tw… — ACT

For the quadratic equation $0.4x^2 + bx - 4.8 = 0$ , where $b$ is a constant, the sum of the two solutions is equal to the product of the two solutions. What is the value of $b$ ?

Answer: 4.8

Answer

The value of the constant

b

4.8

According to Vieta's formulas, the sum of the solutions to the quadratic equation

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

and their product is given by

\frac{c}{a}

. For the given equation

0.4x^2 + bx - 4.8 = 0

, the product of the solutions is

\frac{-4.8}{0.4} = -12

. Setting the sum equal to the product yields the equation

-\frac{b}{0.4} = -12

. Multiplying both sides by

-0.4

isolates

b

, giving

b = 4.8

Step-by-Step Solution

Identify the coefficients of the quadratic equation.

a = 0.4

b = b

, and

c = -4.8

To apply formulas relating the coefficients to the solutions.

Express the sum and product of the solutions using Vieta's formulas.

Sum of solutions is

-\frac{b}{0.4}

and product of solutions is

\frac{-4.8}{0.4} = -12

To establish the mathematical relationship given in the problem.

Equate the sum and product of the solutions and solve for the constant

b

-\frac{b}{0.4} = -12 \implies b = -12 \times (-0.4) = 4.8

The problem states that the sum of the two solutions is equal to their product.

Key Concept

Sum and Product of Roots (Vieta's Formulas)

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