The discriminant, denoted by D (or Delta), is calculated using the formula D = b² - 4ac. This value determines the nature of the roots of the quadratic equation.

Find the discriminant of the quadratic equation $2x^2 - 4x + 3 = 0$ , and hence find the nature of its roots.

Answer: The discriminant is -8. Since the discriminant is less than 0, the quadratic equation has no real roots.

Step-by-step solution

Step 1: Identify coefficients

We compare the given quadratic equation $2x^2 - 4x + 3 = 0$ with the standard form of a quadratic equation, $ax^2 + bx + c = 0$ , to identify the coefficients $a$ , $b$ , and $c$ .

Step 2: Calculate the discriminant

The discriminant, denoted by $D$ (or $\Delta$ ), is calculated using the formula $D = b^2 - 4ac$ . This value determines the nature of the roots of the quadratic equation.

Step 3: Substitute and evaluate

Substitute the values of $a=2$ , $b=-4$ , and $c=3$ into the discriminant formula. First, calculate the square of $b$ , then the product of $4ac$ , and finally subtract the two results to find the value of $D$ .

Step 4: Determine the nature of roots

Since the calculated discriminant $D = -8$ is less than zero ( $D < 0$ ), the quadratic equation has no real roots. This means the roots are complex or imaginary.

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Find the discriminant of the quadratic equation 2x2−4x+3=02x^2 - 4x + 3 = 02x2−4x+3=0, and hence find the nature of its roots.