If m and n are the two distinct real solutions to the quadratic equation 2x^2 - 8x + c = 0… — GRE General Test

If $m$ and $n$ are the two distinct real solutions to the quadratic equation $2x^2 - 8x + c = 0$ , where $c$ is a constant, and $m^2 + n^2 = 12$ , what is the value of $c$ ?

A
-4
4Cevap
C
8
D
2
E
26

Cevap

The value of

c

4

By Vieta's formulas for the quadratic equation

2x^2 - 8x + c = 0

, the sum of the roots is

m + n = 4

and the product of the roots is

mn = c/2

. Squaring the sum gives

(m + n)^2 = m^2 + 2mn + n^2 = 16

. Substituting

m^2 + n^2 = 12

gives

12 + 2(c/2) = 16

, which simplifies to

12 + c = 16

, so

c = 4

Adım Adım Çözüm

Apply Vieta's formulas to determine the sum and product of the roots in terms of the given quadratic coefficients.

For

2x^2 - 8x + c = 0

, the sum of roots is

m + n = -\frac{-8}{2} = 4

, and the product of roots is

m \cdot n = \frac{c}{2}

For a quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and the product of roots is

c/a

Relate

m^2 + n^2

to the sum and product of roots using the identity

(m + n)^2 = m^2 + n^2 + 2mn

m^2 + n^2 = (m + n)^2 - 2mn

Rewriting the sum of squares in terms of

(m+n)

and

mn

allows substitution of the known values.

Substitute

m + n = 4

mn = \frac{c}{2}

, and

m^2 + n^2 = 12

into the equation and solve for

c

12 = (4)^2 - 2\left(\frac{c}{2}\right) \implies 12 = 16 - c \implies c = 4

Direct algebraic simplification yields the constant

c

Anahtar Kavram

Relationship between roots and coefficients of a quadratic equation (Vieta's Formulas) and algebraic identities.

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