The problem asks us to determine the type of solutions the quadratic equation $3x^2 + 10x + 7 = 0$ has, without solving the equation. We are instructed to use the discriminant to do this.

1. Problem Description

The problem asks us to determine the type of solutions the quadratic equation $3x^2 + 10x + 7 = 0$ has, without solving the equation. We are instructed to use the discriminant to do this.

2. Solution Steps

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula:

$D = b^2 - 4ac$

In our case, $a = 3$, $b = 10$, and $c = 7$.

We calculate the discriminant:

$D = (10)^2 - 4(3)(7)$

$D = 100 - 84$

$D = 16$

Since the discriminant $D > 0$, the quadratic equation has two real solutions.

Since $D = 16$ is a perfect square, the solutions are rational.

If $D > 0$ and $D$ is a perfect square, then the equation has two distinct rational solutions.

If $D > 0$ and $D$ is not a perfect square, then the equation has two distinct irrational solutions.

If $D = 0$, then the equation has one rational solution.

If $D < 0$, then the equation has two non-real complex solutions.

In this case, $D = 16 > 0$ and $16$ is a perfect square, so the equation has two rational solutions.

3. Final Answer

Two rational solutions