In the quadratic equation 3x^2 - kx + 24 = 0, k is a positive constant. If one of the root… — SAT

In the quadratic equation $3x^2 - kx + 24 = 0$ , $k$ is a positive constant. If one of the roots of the equation is twice the other root, what is the value of $k$ ?

A
6
B
12
18Cevap
D
36

Cevap

The correct answer is 18. Let the roots of the equation be

r

and

2r

. The product of the roots is given by

\frac{c}{a} = \frac{24}{3} = 8

. Therefore,

2r^2 = 8

, which means

r^2 = 4

. Since

k > 0

, the roots must be positive, so

r = 2

and the two roots are 2 and 4. The sum of the roots is

2 + 4 = 6

. According to Vieta's formulas, the sum of the roots is also equal to

-\frac{b}{a} = \frac{k}{3}

. Setting the two expressions for the sum of the roots equal to each other gives

\frac{k}{3} = 6

, which simplifies to

k = 18

Adım Adım Çözüm

Define the roots and set up the product of the roots using Vieta's formulas.

Let the roots be

r

and

2r

. The product of the roots is

r \times 2r = 2r^2

. From the equation

3x^2 - kx + 24 = 0

, the product of the roots is also

\frac{c}{a} = \frac{24}{3} = 8

This establishes a relationship between the given ratio of the roots and the coefficients of the equation.

Solve for the root variable

r

Setting

2r^2 = 8

gives

r^2 = 4

. Since

k

is a positive constant, the sum of the roots must be positive, which means the roots themselves must be positive. Thus,

r = 2

, and the roots are

2

and

4

Finding the actual values of the roots is necessary to calculate their sum.

Use Vieta's formula for the sum of the roots to find

k

The sum of the roots is

2 + 4 = 6

. According to Vieta's formulas, the sum of the roots is

-\frac{b}{a} = \frac{k}{3}

. Setting them equal gives

\frac{k}{3} = 6

, which simplifies to

k = 18

This directly isolates and solves for the unknown constant

k

Anahtar Kavram

Using Vieta's formulas to relate the roots of a quadratic equation

ax^2 + bx + c = 0

to its coefficients: the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

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