The roots of the quadratic polynomial P(x) = x^2 + bx + c are r_1 and r_2, where b and c a… — GMAT

The roots of the quadratic polynomial $P(x) = x^2 + bx + c$ are $r_1$ and $r_2$ , where $b$ and $c$ are constants. If $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 10x + 21 = 0$ , what is the value of $c$ ?

Answer: 5

Answer

The value of

c

is 5.

Factoring

x^2 - 10x + 21 = 0

yields roots

3

and

7

. Because these roots represent

r_1 + 2

and

r_2 + 2

, subtracting

2

from each root gives the original roots

r_1 = 1

and

r_2 = 5

. In the monic polynomial

P(x) = x^2 + bx + c

, the constant coefficient

c

equals the product of the roots

r_1 \times r_2 = 1 \times 5 = 5

Step-by-Step Solution

Solve for the roots of the given equation

x^2 - 10x + 21 = 0

The roots are

3

and

7

Factoring the quadratic expression yields

(x - 3)(x - 7) = 0

, so

x = 3

x = 7

Determine the roots

r_1

and

r_2

P(x)

r_1 = 1

and

r_2 = 5

Since the roots of the second equation are shifted by

+2

, we set

r_1 + 2 = 3 \implies r_1 = 1

and

r_2 + 2 = 7 \implies r_2 = 5

Calculate the constant term

c

c = 5

By Vieta's formulas, for any monic quadratic polynomial

x^2 + bx + c

, the constant term

c

is the product of the roots

r_1 r_2 = 1 \times 5 = 5

Key Concept

Root transformation of quadratic equations and relationship between roots and coefficients via Vieta's formulas.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic