Find Roots of 2x² − 7x + 3 = 0 — Quadratic Formula Method

Question

Find the roots of: 2x² − 7x + 3 = 0

Solution — Step by Step

This equation has a = 2 (not 1), which makes mental factoring trickier. We'll use the quadratic formula — the method that works every time, no guesswork.

Step 1: Identify a, b, and c.

Compare 2x² − 7x + 3 = 0 with the standard form ax² + bx + c = 0:

• a = 2
• b = −7
• c = 3

Be careful: b is negative. Write it down explicitly before computing.

Step 2: Calculate the discriminant D = b² − 4ac.

We calculate D first because it tells us the nature of roots and determines whether to continue.

D = (−7)² − 4 × 2 × 3 D = 49 − 24 D = 25

Since D = 25 > 0, we have two distinct real roots. Since D is a perfect square, the roots will be rational — a good sign.

Step 3: Find √D.

√25 = 5

Step 4: Apply the quadratic formula.

x = (−(−7) ± 5) / (2 × 2) x = (7 ± 5) / 4

Step 5: Compute both roots.

x₁ = (7 + 5) / 4 = 12/4 = 3

x₂ = (7 − 5) / 4 = 2/4 = 1/2

Verification:

• x = 3: 2(9) − 7(3) + 3 = 18 − 21 + 3 = 0 ✓
• x = 1/2: 2(1/4) − 7(1/2) + 3 = 1/2 − 7/2 + 3 = −3 + 3 = 0 ✓

Why This Works

The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. It works for any values of a, b, c (with a ≠ 0), regardless of whether the roots are integers, fractions, or irrational numbers.

The ± symbol is why we get two roots. Adding √D gives one root, subtracting gives the other. When D = 0, both calculations give the same value (a repeated root).

Alternative Method: Factoring

Let's verify by factoring. For 2x² − 7x + 3:

We need two numbers that multiply to ac = 2 × 3 = 6 and add to b = −7. That's −6 and −1.

2x² − 6x − x + 3 = 0 2x(x − 3) − 1(x − 3) = 0 (2x − 1)(x − 3) = 0

x = 1/2 or x = 3. Same answer.

Common Mistake