How to Use the Discriminant to Count Solutions

Step-by-step solution

1. Read a, b and c.

   $$a = 2,\quad b = -4,\quad c = 5$$

   Why: The coefficients come from ax^2 + bx + c = 0.

2. Substitute into the discriminant.

   $$D = (-4)^2 - 4(2)(5) = 16 - 40 = -24$$

   Why: The discriminant is the expression under the square root in the quadratic formula.

3. Read the sign.

   $$D < 0 \Rightarrow \text{no real solutions}$$

   Why: A negative number has no real square root, so the graph never crosses the x-axis.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $x^2 + 2x + 5 = 0$

   Show answer

   $\text{no real solutions}$

2. 2

   $x^2 - 6x + 9 = 0$

   Show answer

   $\text{one real solution}$

3. 3

   $x^2 - 5x + 6 = 0$

   Show answer

   $\text{two real solutions}$

4. 4

   $x^2 + x + 1 = 0$

   Show answer

   $\text{no real solutions}$

5. 5

   $x^2 - 4x + 1 = 0$

   Show answer

   $\text{two real solutions}$