The system of equations consists of a quadratic equation and a linear equation, where c is… — SAT

The system of equations consists of a quadratic equation and a linear equation, where

c

is a constant:

\begin{aligned} y &= 2x^2 - 3x + 1 \\ y &= x + c \end{aligned}

If this system has exactly one real solution, what is the value of

c

The value of the constant

c

-1

To find the value of

c

for which the system has exactly one real solution, we equate the two equations to get

2x^2 - 3x + 1 = x + c

. Subtracting

x

and

c

from both sides gives the quadratic equation

2x^2 - 4x + (1 - c) = 0

. A quadratic equation has exactly one real solution when its discriminant,

b^2 - 4ac

, is equal to zero. Substituting

a = 2

b = -4

, and the constant term

(1 - c)

into the discriminant formula gives

(-4)^2 - 4(2)(1 - c) = 0

, which simplifies to

16 - 8(1 - c) = 0

. Solving for

c

yields

8 + 8c = 0

, or

c = -1

Equate the two expressions for

y

to set up a single equation in terms of

x

2x^2 - 3x + 1 = x + c

Since both equations are equal to

y

, their right-hand sides must be equal at any point of intersection.

Rearrange the equation into standard quadratic form,

ax^2 + bx + c_0 = 0

2x^2 - 4x + (1 - c) = 0

Subtracting

x

and

c

from both sides groups the terms by their powers of

x

Write the expression for the discriminant of this quadratic equation.

D = (-4)^2 - 4(2)(1 - c)

For a quadratic equation

ax^2 + bx + c_0 = 0

, the discriminant is given by

b^2 - 4ac_0

where

a = 2

b = -4

, and the constant term is

1 - c

Set the discriminant to zero and solve for

c

to find the condition for exactly one real solution.

16 - 8(1 - c) = 0 \implies 16 - 8 + 8c = 0 \implies 8 + 8c = 0 \implies c = -1

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

Determining the number of solutions of a nonlinear system by setting the discriminant of the combined quadratic equation to zero.

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