Math Snap

STEP 1

Assumptions1. The two equations given are quadratic equations.
. A quadratic equation $ax^ + bx + c =0$ has real roots if the discriminant $b^ -4ac$ is greater than or equal to zero.
3. We need to find the range of values of $p$ for which both equations have real roots.

STEP 2

Let's start with the first equation. The discriminant of the first equation is $1 = (2p)^2 -4**$.

STEP 3

implify the discriminant of the first equation.
$$1 =p^2 -36$$

STEP 4

For the first equation to have real roots, the discriminant $1$ must be greater than or equal to zero. So, we have the inequality$$4p^2 -36 \geq0$$

STEP 5

olve the inequality for $p$.
$$4p^2 \geq36$$$$p^2 \geq9$$$$p \geq3 \quad or \quad p \leq -3$$

STEP 6

Now, let's move to the second equation. The discriminant of the second equation is $2 = (4)^2 -4<em>1</em>(-p)$.

STEP 7

implify the discriminant of the second equation.
$$2 =16 +4p$$

STEP 8

For the second equation to have real roots, the discriminant $2$ must be greater than or equal to zero. So, we have the inequality$$16 +4p \geq0$$

STEP 9

olve the inequality for $p$.
$$4p \geq -16$$$$p \geq -4$$

The range of $p$ for which both equations have real roots is the intersection of the ranges found in steps5 and9. So, we have$$p \geq3 \quad or \quad -4 \leq p \leq -3$$The range of values of $p$ for which both the equations $3 x^{2}+2 p x+3=0$ and $x^{2}+4 x-p=0$ have real roots is $p \geq3$ and $-4 \leq p \leq -3$.