Solve 3x² + 10x + 3 = 0

Solve the quadratic equation 3x² + 10x + 3 = 0 using the quadratic formula.

Solution:

Discriminant: Δ = 64 (2 real roots)

Step-by-Step Solution

Step 1 Identify the Standard Form

A quadratic equation has the form: ax² + bx + c = 0

From the given equation:

• a = 3 (coefficient of x²)

• b = 10 (coefficient of x)

• c = 3 (constant term)

Standard form: 3x² + 10x + 3 = 0

3x² + 10x + 3 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (10)² - 4(3)(3)

D = 100 - 36

D = 64

Since D = 64 > 0 and √D = 8 (perfect square),

the equation has two distinct RATIONAL roots.

D = 64

Step 3 Solve by Factoring (AC Method)

For 3x² + 10x + 3 = 0:

Step 1: Find ac = 3 × 3 = 9

Step 2: Find factors of 9 that sum to 10

The equation factors to:

(3x + 1)(x + 3) = 0

\text{AC Method}

Step 4 Find the Solutions

Setting each factor to zero:

x₁ = -0.333333

x₂ = -3

x_1 = -0.333333, x_2 = -3

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = -0.333333:

3(-0.333333)² + 10(-0.333333) + 3

= 0 ✓

For x = -3:

3(-3)² + 10(-3) + 3

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -3 or x = -0.333333