Mathematics 10 — Exercise 2.3 · Q4 (iii)

Full question

If α,β are roots of x² + 2x + 4 = 0, form the equation with the displayed new roots.

Step-by-step solution

1. For the original monic equation, sum of roots = −linear coefficient and product = constant.

   $$(\alpha +\beta =-2,\alpha \times \beta =4)$$

2. Add the two required roots to obtain their new sum.

   $$\mathrm{Sum}=-\frac{7}{2}$$

3. Multiply the two required roots to obtain their new product.

   $$\mathrm{Product}=\frac{49}{4}$$

4. A monic quadratic with roots r₁,r₂ is x² − (sum)x + product = 0. Substitute the two values.

   $$x^2+\frac{7\times x}{2}+\frac{49}{4}=0$$

5. Multiply through by the constant common denominator to obtain integer coefficients.

   $$4\times x^2+14\times x+49=0$$

Answer

The final answer is shown below.