Find the roots of the equation 2x2 – 5x + 3 = 0, by factorisation.
Step-by-Step Solution
Key Concept: Factorise the quadratic expression into two linear factors of the form \((ax + b)(cx + d) = 0\). The values of \(x\) that make each factor zero are the roots of the equation.
1. Write the quadratic in standard form
$$2x^{2} - 5x + 3 = 0.$$
2. Assume a factorisation
Since the leading coefficient is 2, try factors of the form \((2x + p)(x + q)\) where \(p\) and \(q\) are integers. Expanding gives
$$ (2x + p)(x + q) = 2x^{2} + (2q + p)x + pq.$$
Compare with \(2x^{2} - 5x + 3\).
3. Match the coefficients
- Coefficient of \(x\): \(2q + p = -5\).
- Constant term: \(pq = 3\).
4. Find integer pairs \((p,q)\) satisfying \(pq = 3\)
The possibilities are \((1,3), (3,1), (-1,-3), (-3,-1)\).
5. Test each pair in the linear‑coefficient equation
- For \(p = -3, q = -1\): \(2(-1) + (-3) = -2 -3 = -5\) ✔️
Hence \(p = -3\) and \(q = -1\) satisfy both conditions.
6. Write the factorised form
$$2x^{2} - 5x + 3 = (2x - 3)(x - 1).$$
7. Apply the zero‑product property
Set each factor to zero:
\[2x - 3 = 0 \quad \text{or} \quad x - 1 = 0.\]
Solving gives
\[x = \frac{3}{2} \quad \text{or} \quad x = 1.\]
8. State the roots
The roots of the quadratic equation are \(x = 1\) and \(x = \dfrac{3}{2}\).
Correct Answer: The roots are \(x = 1\) and \(x = \dfrac{3}{2}\).