The quadratic equation x^2 - 4x + 2 = 0 has roots r_1 and r_2. A second quadratic equation… — SAT

The quadratic equation $x^2 - 4x + 2 = 0$ has roots $r_1$ and $r_2$ . A second quadratic equation, $x^2 - px + q = 0$ , has roots $\left(r_1 + \frac{1}{r_2}\right)$ and $\left(r_2 + \frac{1}{r_1}\right)$ , where $p$ and $q$ are constants. What is the value of $p$ ?

Answer: 6

Answer

The value of

p

is 6.

By Vieta's formulas, the sum of the roots of

x^2 - 4x + 2 = 0

r_1 + r_2 = 4

and the product of the roots is

r_1 r_2 = 2

. The sum of the roots of the second equation

x^2 - px + q = 0

p

. Therefore,

p = \left(r_1 + \frac{1}{r_2}\right) + \left(r_2 + \frac{1}{r_1}\right) = (r_1 + r_2) + \frac{r_1 + r_2}{r_1 r_2}

. Substituting the known values yields

p = 4 + \frac{4}{2} = 6

Step-by-Step Solution

Determine the sum and product of the roots of the first equation.

r_1 + r_2 = 4

and

r_1 r_2 = 2

Vieta's formulas state that for

x^2 - Bx + C = 0

, the sum of the roots is

B

and the product is

C

Express the sum of the roots of the second equation, which is

p

p = (r_1 + r_2) + \left(\frac{1}{r_1} + \frac{1}{r_2}\right)

The sum of the roots of

x^2 - px + q = 0

p

Simplify the fractional part of the equation.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

Finding a common denominator allows us to write the sum of reciprocals in terms of the sum and product of the roots.

Substitute the values of the sum and product into the expression for

p

p = 4 + \frac{4}{2} = 6

This evaluates the expression to find the final value of

p

Key Concept

Relating the roots and coefficients of quadratic equations using Vieta's formulas.

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