Question

How to use the discriminant to find out how many real number roots an equation has for ${{x}^{2}}-8x+3=0$?

Answer 1

Hint: We know that the solution of a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ can be find by using the quadratic formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . The discriminant is the portion of quadratic formula which is given as $D={{b}^{2}}-4ac$. The nature of roots depends on the value of discriminate so by finding the value of discriminant for the given equation we will find the nature of roots.

Complete step by step solution:
We have been given an equation ${{x}^{2}}-8x+3=0$.
We have to find the number of real roots an equation has.
Now, we know that the discriminant is the portion of the quadratic formula which is given by $D={{b}^{2}}-4ac$.
Now, we know that the nature of roots of a quadratic equation depends on the value of discriminate. So let us first calculate the value of discriminant for the given equation.
By comparing the given equation with the general equation $a{{x}^{2}}+bx+c=0$ we get the values as
$\Rightarrow a=1,b=-8,c=3$
Now, substituting the values in the discriminant formula we will get
$\Rightarrow D={{\left( -8 \right)}^{2}}-4\times 1\times 3$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow D=64-12 \\
 & \Rightarrow D=52 \\
\end{align}$
Therefore we get the positive and real value of D.
Hence the given equation has two real and positive roots.

Note: The point to be noted is that if the value of discriminant is zero the equation has only one solution, if the value of discriminant is negative then the equation has complex roots. The number of roots or solutions also depends on the degree of the equation.