Question

What is the nature of the roots of the quadratic equation $x^2 - 7x + 10 = 0$?

• A. Real and distinct roots
• B. Real and equal roots
• C. No real roots
• D. None of the above

Hint:

A positive discriminant always means two different real roots. \[ D>0 \Rightarrow \text{Real and Distinct} \] \[ D=0 \Rightarrow \text{Real and Equal} \] \[ D<0 \Rightarrow \text{No Real Roots} \]

Answer 1

The Correct Option is A

Concept: The nature of roots of a quadratic equation depends on the discriminant. For the equation \[ ax^2+bx+c=0 \] the discriminant is: \[ D=b^2-4ac \] Nature of roots based on discriminant:
• If $D>0$, roots are real and distinct.
• If $D=0$, roots are real and equal.
• If $D<0$, roots are imaginary or non-real.

Step 1: Identify the coefficients. Given equation: \[ x^2 - 7x + 10 = 0 \] Comparing with \[ ax^2 + bx + c = 0 \] we get: \[ a = 1 \] \[ b = -7 \] \[ c = 10 \]

Step 2: Calculate the discriminant. \[ D = b^2 - 4ac \] Substitute the values: \[ D = (-7)^2 - 4(1)(10) \] \[ D = 49 - 40 \] \[ D = 9 \]

Step 3: Interpret the value of discriminant. Since \[ D = 9 > 0 \] the roots are: \[ \boxed{\text{Real and distinct}} \] Hence, the correct option is: \[ \boxed{(1)\ \text{Real and distinct roots}} \]