Question

Let N be the number of quadratic equations with coefficients from {0,1,2,...,9} such that 0 is a solution of each equation. Then the value of N is:

• A. 29
• B. 39
• C. 90
• D. 81

Hint:

When solving for the number of solutions of a quadratic equation with a specific root, substitute the root into the equation and simplify. Then, count the number of valid combinations for the coefficients.

Answer 1

The Correct Option is C

Step 1: The general form of a quadratic equation is:

\[ ax^2 + bx + c = 0 \]

For 0 to be a root, substitute \( x = 0 \) into the equation:

\[ a(0)^2 + b(0) + c = 0 \implies c = 0. \]

Therefore, the quadratic equation simplifies to:

\[ ax^2 + bx = 0 \]

Step 2: Factor the equation:

\[ x(ax + b) = 0 \]

This shows that for 0 to be a root, the equation must be of the form \( x(ax + b) = 0 \).

Step 3: The coefficients \( a \) and \( b \) can each take values from the set \( \{0, 1, 2, \ldots, 9\} \), with the restriction that \( a \neq 0 \) (since it is a quadratic equation).

Step 4: The number of possible values for \( a \) is 9 (since \( a \in \{1, 2, \ldots, 9\} \)) and the number of possible values for \( b \) is 10 (since \( b \in \{0, 1, 2, \ldots, 9\} \)).

Step 5: Therefore, the total number of quadratic equations where 0 is a root is:

\[ 9 \times 10 = 90. \]