Question

In the following, determine the set of values of k for which the given quadratic equation has real roots:
$\left(i\right)2x^2+3x+k=0\left(ii\right)2x^2+x+k=0\left(iii\right)2x^2-5x-k=0\left(iv\right)k{x}^2+6x+1=0\left(v\right)3x^2+2x+k=0$

Answer 1

For the quadratic equation $a{x}^2+bx+c=0$, roots are real if $b^2-4ac\ge 0$

(i) $3^2-4\times 2\times k\ge 0$

$\Rightarrow k\le \frac{9}{8}$

(ii) $1^2-4\times 2k\ge 0$

$\Rightarrow k\le \frac{1}{8}$

(iii) $(-5{)}^2-4\times 2\times (-k)\ge 0$

$\Rightarrow k\ge -\frac{25}{8}$

(iv) $6^2-4k\ge 0$

$\Rightarrow k\le 9$

(v) $2^2-4\times 3\times k\ge 0$

$\Rightarrow k\le \frac{1}{3}$