A quadratic equation of the form a x^2 + b x + c = 0, where a, b, and c are real constants… — ACT

A quadratic equation of the form $a x^2 + b x + c = 0$ , where $a$ , $b$ , and $c$ are real constants and $a > 0$ , has a discriminant of $37$ . If the sum of the roots of this equation is $5.5$ and the product of the roots is $5.25$ , what is the value of the coefficient $a$ ?

Answer: 2

Answer

The value of the coefficient

a

2

By applying Vieta's formulas, we can write

b = -5.5a

and

c = 5.25a

. Plugging these into the discriminant formula gives

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

. Setting the discriminant to

37

results in

9.25a^2 = 37 \implies a^2 = 4

. Since the problem specifies

a > 0

, taking the positive square root gives

a = 2

Step-by-Step Solution

Express the coefficients

b

and

c

in terms of

a

using Vieta's formulas.

b = -5.5a

and

c = 5.25a

The sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

Substitute the expressions for

b

and

c

into the discriminant formula

D = b^2 - 4ac

D = 9.25a^2

Substituting the terms yields

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

Equate the discriminant expression to

37

and solve for

a

a = 2

Since

D = 37

, we write

9.25a^2 = 37 \implies a^2 = 4

. Because

a

must be positive, we find

a = 2

Key Concept

Using the properties of quadratic roots (Vieta's formulas) and the definition of the discriminant to solve for coefficients.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic