Question4) $3 p^{2}+8 p+4=0$

Studdy Solution

STEP 1

1. The equation $3p^2 + 8p + 4 = 0$ is a quadratic equation.
2. The goal is to find the values of $p$ that satisfy the equation.

STEP 2

1. Identify the coefficients of the quadratic equation.
2. Use the quadratic formula to find the roots.
3. Simplify the expression to find the values of $p$ .

STEP 3

Identify the coefficients from the quadratic equation $3p^2 + 8p + 4 = 0$ . The standard form of a quadratic equation is $ax^2 + bx + c = 0$ , where: - $a = 3$ - $b = 8$ - $c = 4$

STEP 4

Apply the quadratic formula, which is given by:
$p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Substitute the identified coefficients $a = 3$ , $b = 8$ , and $c = 4$ into the formula.

STEP 5

Calculate the discriminant $b^2 - 4ac$ :
$b^2 - 4ac = 8^2 - 4 \times 3 \times 4$ $= 64 - 48$ $= 16$

STEP 6

Substitute the discriminant back into the quadratic formula:
$p = \frac{-8 \pm \sqrt{16}}{2 \times 3}$

STEP 7

Simplify the expression under the square root and solve for $p$ :
$p = \frac{-8 \pm 4}{6}$
This yields two potential solutions for $p$ :
$p = \frac{-8 + 4}{6} \quad \text{and} \quad p = \frac{-8 - 4}{6}$

STEP 8

Calculate each solution:
For $p = \frac{-8 + 4}{6}$ :
$p = \frac{-4}{6} = -\frac{2}{3}$
For $p = \frac{-8 - 4}{6}$ :
$p = \frac{-12}{6} = -2$
The solutions for $p$ are:
$p = -\frac{2}{3} \quad \text{and} \quad p = -2$

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