Question:
<p>The zeroes of the polynomial x<sup>2</sup> - 3x - m (m + 3) are:</p>
<p style="display:inline">m, m + 3</p>
<p style="display:inline">–m, m + 3</p>
<p style="display:inline">m, –(m + 3)</p>
<p style="display:inline">–m, –(m + 3)</p>
Step-by-Step Solution
Key Concept: Solve the quadratic equation by applying the quadratic formula and simplifying the discriminant into a perfect square of a binomial involving the parameter m.
<p>Given: equation x<sup>2 </sup>− 3x − m(m + 3) = 0, where m is a constant<br />
The given equation is the form of ax<sup>2</sup> + bx + c = 0<br />
<span class="math-tex">$\therefore$</span> a = 1, b = −3, c = −m(m + 3)<br />
We know the roots of the equation can be find out using the formula,<br />
<span class="math-tex">$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$</span><br />
Substituting the values of a, b, c, we get<br />
<span class="math-tex">$x=\frac{-(-3) \pm \sqrt{(-3)^{2}-4(1)(-m(m+3)}}{2}$</span><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$x=\frac{3 \pm \sqrt{9+4 m^{2}+12 m}}{2}$</span><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$x=\frac{3 \pm(2 m+3)}{2}$</span><br />
or <span class="math-tex">$x=\frac{3+(2 m+3)}{2}$</span>, <span class="math-tex">$x=\frac{3-(2 m+3)}{2}$</span><br />
<span class="math-tex">$\Rightarrow$</span> x = m + 3 and x = −m are the required roots of the equation.</p>
Correct Answer: B