Question

Write the discriminant of the following equation: \[{x^2} - 4x + 2 = 0\].

Answer 1

Hint:
Here in this question, the equation given to us is a quadratic equation. We will compare the given quadratic equation with the standard form of a quadratic equation \[a{x^2} + bx + c = 0\] to find the coefficients \[a\], \[b\], and \[c\]. We will substitute the values of the coefficients in the formula \[D = {b^2} - 4ac\] to find the discriminant.

Complete step by step solution:
We know that a quadratic equation is an equation where 2 is the highest power of the variable \[x\].
The standard form of a quadratic equation is \[a{x^2} + bx + c = 0\], where \[a\], \[b\], and \[c\] are real numbers, and \[a\] is not equal to 0.
The discriminant of a quadratic equation of the form \[a{x^2} + bx + c = 0\] is given by \[D = {b^2} - 4ac\].
A discriminant helps to identify the nature of roots of a quadratic equation.
The roots of a quadratic equation are real and distinct when the discriminant is \[D > 0\], the roots are real and equal when the discriminant is \[D = 0\], and the roots are complex when the discriminant is \[D < 0\].
Now, we will compare the given quadratic equation with the standard form of a quadratic equation, that is compare \[{x^2} - 4x + 2 = 0\] with \[a{x^2} + bx + c = 0\].
Comparing the coefficients of the variables in the two equations, we can identify the values of the coefficients.
\[a = 1\]
\[b = - 4\]
\[c = 2\]
Next, we will substitute 1 for \[a\], \[ - 4\] for \[b\] and 2 for \[c\] in the formula for discriminant of a quadratic equation.
Thus, we get
\[D = {b^2} - 4ac\\ = {\left( { - 4} \right)^2} - 4\left( 1 \right)\left( 2 \right)\\ = 16 - 8\\ = 8\]

Therefore, the discriminant of the equation \[{x^2} - 4x + 2 = 0\] is 8.

Note:
Quadratic equations are the equations in which the highest power of a variable is 2. The solutions of these equations are based on their discriminants. The discriminant shows the nature of the roots of the quadratic equation. The value of the discriminant tells us whether the roots are real and equal, real and unequal or imaginary.