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: A quadratic equation's standard form is:

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

If 0 is a root, substitute \( x = 0 \):

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

Thus, the equation simplifies to:

\[ ax^2 + bx = 0 \]

Step 2: Factor the equation:

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

For 0 to be a root, the equation must be \( x(ax + b) = 0 \).

Step 3: The coefficients \( a \) and \( b \) can be any value from \( \{0, 1, 2, \ldots, 9\} \), with \( a \neq 0 \) (for a quadratic equation).

Step 4: There are 9 possible values for \( a \) (since \( a \in \{1, 2, \ldots, 9\} \)) and 10 for \( b \) (since \( b \in \{0, 1, 2, \ldots, 9\} \)).

Step 5: Consequently, the total number of quadratic equations with 0 as a root is:

\[ 9 \times 10 = 90. \]