Question

Find the nature of the roots of 4x^2 + 10x + 6 = 0.

Answer 1

Rational

Explanation

Step 1

to find the nature of the roots of the quadratic equation, we will first calculate the roots of the equation.

$4x^2+10x+6=0$

to solve for x, we can use the quadratic formula

remember:

$\begin{gathered} \text{ for} \\ ax^2+bx+c=0 \\ th\text{e solutino for x is} \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{gathered}$

now, the discriminant is

$b^2-4ac$

so, to check the nature of the roots we have these criteria

$\begin{gathered} b^2-4ac>0\Rightarrow2\text{ distintc real roots} \\ b^2-4ac=0\Rightarrow one\text{ repeated real root} \\ b^2-4ac<0\Rightarrow complex\text{ root} \end{gathered}$

so, let

$\begin{gathered} ax^2+bx+c==\Rightarrow4x^2+10x+6=0 \\ so \\ a=4 \\ b=10 \\ c=6 \end{gathered}$

now, replace in the discriminat formula

$\begin{gathered} b^2-4ac \\ (10^2)-4(4)(6) \\ 100-96 \\ 4 \end{gathered}$

so, the discrimant is 4, therefore

$\begin{gathered} b^{2}-4ac\gt0 \\ 4>0\Rightarrow \end{gathered}$

Rational

I hope this helps you