Solve 6x² + 7x − 3 = 0 by factorisation.

1/3 and -3/2. ac = 6 × (−3) = −18, and 9 × (−2) = −18 with 9 + (−2) = 7. So 6x² + 9x − 2x − 3 = 3x(2x + 3) − 1(2x + 3) = (2x + 3)(3x − 1) = 0, giving x = 1/3 or x = -3/2. Check 1/3: 6/9 + 7/3 - 3 = 0. The other three options each flip one or both signs — a factor (3x − 1) gives a POSITIVE root and (2x + 3) a NEGATIVE one, not the reverse.

Solve $6 x^{2} + 7 x - 3 = 0$ by factorisation.

Class 10Quadratic EquationsAlgebra — Quadratics

A $- \frac{1}{3}$ and $\frac{3}{2}$
B $\frac{1}{3}$ and $- \frac{3}{2}$
C $\frac{1}{3}$ and $\frac{3}{2}$
D $- \frac{1}{3}$ and $- \frac{3}{2}$

Step-by-Step Solution

ac =

6 \times (- 3) = - 18

, and

9 \times (- 2) = - 18

with

9 + (- 2) = 7

. So

6 x^{2} + 9 x - 2 x - 3 = 3 x (2 x + 3) - 1 (2 x + 3) = (2 x + 3) (3 x - 1) = 0

, giving

x = \frac{1}{3}

x = - \frac{3}{2}

. Check

\frac{1}{3} : \frac{6}{9} + \frac{7}{3} - 3 = 0

. The other three options each flip one or both signs — a factor

(3 x - 1)

gives a POSITIVE root and

(2 x + 3)

a NEGATIVE one, not the reverse.

Answer: B)

\frac{1}{3}

and

- \frac{3}{2}

Key idea: Method 1: Factorisation

Rewrite

a x^{2} + b x + c = 0

as a product

(p x + q) (r x + s) = 0

, then use the **zero-product property**: a product is zero iff one of its factors is zero.

**Steps for

x^{2} + b x + c = 0 * * (a = 1)

:
1. Find two numbers that MULTIPLY to

c

and ADD to

b

.
2. Use them as the constants in

(x + p) (x + q)

.

**Example**:

x^{2} - 7 x + 12 = 0

→ product 12, sum −7 → numbers −3 and −4 →

(x - 3) (x - 4) = 0

→

x = 3

or 4.

**For

a x^{2} + b x + c

with a ≠ 1**: split the middle term — find two numbers multiplying to

a \times c

and summing to

b

. Then group.

Example:

2 x^{2} + 7 x + 3 = 0. a \times c = 6

, sum = 7 → 6 and 1. Split:

2 x^{2} + 6 x + x + 3 = 2 x (x + 3) + 1 (x + 3) = (2 x + 1) (x + 3) = 0

→

x = - 1/2

or −3.

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Solve 6x2+7x−3=0 by factorisation.

Step-by-Step Solution

Key idea: Method 1: Factorisation

Master this chapter in 30 minutes a day.

More Class 10 Quadratic Equations practice

Solve $6 x^{2} + 7 x - 3 = 0$ by factorisation.