Question

How do you find the discriminant of $5{x^2} - 4x + 1 = 3x$ and use it to determine if the equation has one, two, real or two imaginary roots?

Answer 1

Hint: According to the question given in the question we have to find the discriminant of $5{x^2} - 4x + 1 = 3x$ and use it to determine if the equation has one, two, real or two imaginary roots. So, to determine if the equation has one, two, real or two imaginary roots first of all we have to rearrange the given expression in the standard form which can be done by taking all the right hand side terms to the left hand side.
Now, we have to determine the roots of the obtained standard form of the expression which can be done by using the formula to determine the roots of the quadratic expression which is as explained below:

Formula used:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}...............(A)$
Hence, a, is the coefficient of ${x^2}$, b is the coefficient of x and c is the constant term.
Now, to discriminant for this portion of the quadratic expression within the radicals $\sqrt {{b^2} - 4ac} $ and if the discriminant is positive so we will obtain two real solutions, if the discriminant is zero we will obtain, just one solution and if the discriminant is negative we will get complex solutions.

Complete step by step solution:
Step 1: First of all we have to rearrange the given expression in the standard form which can be done by taking all the right hand side terms to the left hand side. Hence,
$
   \Rightarrow 5{x^2} - 4x + 1 - 3x = 0 \\
   \Rightarrow 5{x^2} - 7x + 1 = 0 \\
 $
Step 2: Now, we have to determine the roots of the obtained standard form of the expression which can be done by using the formula (A) to determine the roots of the quadratic expression which is as explained in the solution hint. Hence,
$
   \Rightarrow {( - 7)^2} - 4 \times 5 \times 1 \\
   \Rightarrow 49 - 20 \\
   \Rightarrow 29 \\
 $
Step 3: Now, to discriminant for this portion of the quadratic expression within the radicals $\sqrt {{b^2} - 4ac} $ and if the discriminant is positive so we will obtain two real solutions, if the discriminant is zero we will obtain, just one solution and if the discriminant is negative we will get complex solutions.
$ \Rightarrow \sqrt {{b^2} - 4ac} = 29$
So, we can say that the discriminant is positive we will obtain two real solutions or roots.

Final solution: Hence, with the help of the formula (A) as mentioned in the solution hint we have determined that the discriminant is positive we will obtain two real solutions or roots.

Note:
1. If the discriminant is positive so we will obtain two real solutions, if the discriminant is zero we will obtain, just one solution and if the discriminant is negative we will get complex solutions.
2. If $\sqrt {{b^2} - 4ac} > 0$ then the roots are real and different, if $\sqrt {{b^2} - 4ac} = 0$ then the roots are real and repeated and if $\sqrt {{b^2} - 4ac} < 0$ then the roots are imaginary.