Polynomials
Grade Class 10
Question:
<p><strong>Assertion (A):</strong> x<sup>2</sup> + 7x + 12 has no real zeros<br />
<strong>Reason (R):</strong> A quadratic polynomial can have at the most two zeroes.</p>
<p style="display:inline">Both A and R are true and R is the correct explanation of A.</p>
<p style="display:inline">Both A and R are true but R is not the correct explanation of A.</p>
<p style="display:inline">A is true but R is false.</p>
<p style="display:inline">A is false but R is true.</p>
Step-by-Step Solution
Key Concept: To determine if a quadratic polynomial has real zeros, verify if the discriminant ($D = b^2 - 4ac$) is non-negative or attempt to factorize the expression into linear terms.
<p>x<sup>2</sup> + 7x + 12 = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> x<sup>2</sup> + 4x + 3x + 12 = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> x(x + 4) + 3(x + 4) = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> (x + 4) (x + 3) = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> (x + 4) = 0 or (x + 3) = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> x = -4 or x = -3<br />
Therefore, x<sup>2</sup> + 7x + 12 has two real zeroes. </p>
Correct Answer: D