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 - 6)(x - 0.5) = 0
\text{AC Method}
Step 4
Find the Solutions
Setting each factor to zero:
x₁ = 3
x₂ = 0.5
x_1 = 3, x_2 = 0.5
Step 5
Verify the Solutions
Substitute each solution back into the original equation:
For x = 3:
2(3)² + -7(3) + 3
= 0 ✓
For x = 0.5:
2(0.5)² + -7(0.5) + 3
= 0 ✓
\text{Both solutions verified!}
Step 6
Final Answer
x = 0.5 or x = 3