Determine if Prime 15x^2+10x-9x+7

$15 x^{2} + 10 x - 9 x + 7$

Step 1

Find the discriminant for $15 x^{2} + 10 x - 9 x + 7 = 0$ . In this case, $b^{2} - 4 a c = - 419$ .

Simplify the left side.

Subtract $9 x$ from $10 x$ .

$15 x^{2} + x + 7 = 0$

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Substitute in the values of $a$ , $b$ , and $c$ .

$1^{2} - 4 (15 \cdot 7)$

Evaluate the result to find the discriminant.

Simplify each term.

One to any power is one.

$1 - 4 (15 \cdot 7)$

Multiply $- 4 (15 \cdot 7)$ .

Multiply $15$ by $7$ .

$1 - 4 \cdot 105$

Multiply $- 4$ by $105$ .

$1 - 420$

Subtract $420$ from $1$ .

$- 419$

Step 2

A perfect square number is an integer that is the square of another integer. $\sqrt{- 419} \approx i \sqrt{419}$ , which is not an integer number.

$\sqrt{- 419} \approx i \sqrt{419}$

Step 3

Since $- 419$ can\'t be the square of another integer, it is not a perfect square number.

Step 4

The polynomial $15 x^{2} + 10 x - 9 x + 7$ is prime because the discriminant is not a perfect square number.

Prime