Use the discriminant of the following quadratic equation to state the number and type of solutions.

[tex]\[ 3x^2 + 4x - 5 = 3x + 2 \][/tex]

A. 2 complex solutions
B. 2 real solutions
C. 1 complex solution
D. 1 real solution



Answer :

To determine the number and type of solutions for the given quadratic equation [tex]\( 3x^2 + 4x - 5 = 3x + 2 \)[/tex], we first need to get it into the standard form [tex]\( Ax^2 + Bx + C = 0 \)[/tex].

1. Start by isolating all terms on one side of the equation:
[tex]\[ 3x^2 + 4x - 5 - (3x + 2) = 0 \][/tex]

2. Simplify the equation by combining like terms:
[tex]\[ 3x^2 + 4x - 5 - 3x - 2 = 0 \][/tex]
[tex]\[ 3x^2 + (4x - 3x) + (-5 - 2) = 0 \][/tex]
[tex]\[ 3x^2 + x - 7 = 0 \][/tex]

Now we have the quadratic equation in the standard form:
[tex]\[ 3x^2 + x - 7 = 0 \][/tex]

To determine the number and type of solutions, we need to calculate the discriminant [tex]\(\Delta\)[/tex], which is given by:
[tex]\[ \Delta = B^2 - 4AC \][/tex]

For the quadratic equation [tex]\(Ax^2 + Bx + C = 0\)[/tex], we identify:
[tex]\[ A = 3, \quad B = 1, \quad C = -7 \][/tex]

Plug these values into the discriminant formula:
[tex]\[ \Delta = 1^2 - 4 \cdot 3 \cdot (-7) \][/tex]
[tex]\[ \Delta = 1 - (-84) \][/tex]
[tex]\[ \Delta = 1 + 84 \][/tex]
[tex]\[ \Delta = 85 \][/tex]

The discriminant [tex]\(\Delta\)[/tex] is 85. Based on the value of the discriminant, the type of solutions can be determined as follows:

- If [tex]\(\Delta > 0\)[/tex]: There are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex]: There is exactly one real solution.
- If [tex]\(\Delta < 0\)[/tex]: There are two complex solutions.

Since [tex]\(\Delta = 85 > 0\)[/tex], there are two real solutions to the equation.

Therefore, the number and type of solutions for the equation [tex]\( 3x^2 + 4x - 5 = 3x + 2 \)[/tex] is:
[tex]\[ 2 \text{ real solutions} \][/tex]