Solve 2x² + 7x + 3 = 0

Solve the quadratic equation 2x² + 7x + 3 = 0 using the quadratic formula.

Solution:

Discriminant: Δ = 25 (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 = 2 (coefficient of x²)

• b = 7 (coefficient of x)

• c = 3 (constant term)

Standard form: 2x² + 7x + 3 = 0

2x² + 7x + 3 = 0

Step 2 Calculate the Discriminant

The discriminant determines the nature of the roots.

D = b² - 4ac

D = (7)² - 4(2)(3)

D = 49 - 24

D = 25

Since D = 25 > 0 and √D = 5 (perfect square),

the equation has two distinct RATIONAL roots.

D = 25

Step 3 Solve by Factoring (AC Method)

For 2x² + 7x + 3 = 0:

Step 1: Find ac = 2 × 3 = 6

Step 2: Find factors of 6 that sum to 7

The equation factors to:

(2x + 1)(x + 3) = 0

\text{AC Method}

Step 4 Find the Solutions

Setting each factor to zero:

x₁ = -0.5

x₂ = -3

x_1 = -0.5, x_2 = -3

Step 5 Verify the Solutions

Substitute each solution back into the original equation:

For x = -0.5:

2(-0.5)² + 7(-0.5) + 3

= 0 ✓

For x = -3:

2(-3)² + 7(-3) + 3

= 0 ✓

\text{Both solutions verified!}

Step 6 Final Answer

x = -3 or x = -0.5