Question

Determine the value of missing coefficient so that the roots of the equation are equal and are both rational
A. 4x^2+20x+_=0
B. _x^2+30x+25=0
C. x^2+_x+9/4=0​

Answer 1

For roots to be equal and rational, D=0

If D=0,then roots are given by $\sf \frac{-b} {2a}</p><p>$

Now.

$A) \:\sf 4x^2+20x+c=0 \\\\\sf</p><p>D=b^2-4ac \\\\\sf</p><p>0=400-16c \\\\\sf</p><p>16c=400 \\\\\sf</p><p>\boxed{c=25} </p><p>$

$B)\: \sf ax^2+30x+25=0 \\\\\sf</p><p>D=b^2-4ac \\\\\sf</p><p>0=900-100a \\\\\sf</p><p>100a=900 \\\\\sf</p><p>\boxed{a=9} \\\\\sf</p><p></p><p>Now, \:Roots\:=\frac{-b} {2a} \\\\\sf</p><p>= \frac{-30}{18} \\\\\sf</p><p>=\frac{-5}{3} </p><p>$

$C) \:\sf x^2+bx+\frac{9}{4}=0 \\\\\sf</p><p>D=b^2-4ac \\\\\sf</p><p>0=b^2-9 \\\\\sf</p><p>b^2=9 \\\\\sf</p><p>\boxed{b=3} \\\\\sf</p><p></p><p>Now, \:Roots\:=\frac{-b} {2a} \\\\\sf</p><p>= \frac{-3}{2} \\\\\sf</p><p>$