Example Card: Finding the Number of Solutions
Determine the number of solutions to a quadratic equation in Grade 9 algebra using the discriminant b²-4ac: positive means two real solutions, zero means one, and negative means no real solutions.
Key Concepts
Let's see how one quick calculation reveals an equation's number of solutions, saving us from the full quadratic formula. This first key idea, using the discriminant, allows us to predict the number of real solutions without fully solving the equation.
Example Problem Use the discriminant to find the number of real solutions for $3x^2 + 5x 2 = 0$.
Step by Step 1. In the equation $3x^2 + 5x 2 = 0$, we identify our coefficients: $a=3$, $b=5$, and $c= 2$. 2. We'll use the discriminant formula, which is the part under the square root in the quadratic formula: $$ b^2 4ac $$ 3. Now, substitute the values for $a$, $b$, and $c$: $$ (5)^2 4(3)( 2) $$ 4. Simplify the calculation: $$ 25 ( 24) = 49 $$ 5. The discriminant is $49$. Because it's a positive number, there are two distinct real solutions. This means the graph of the function will have two x intercepts.
Common Questions
What is the discriminant and how does it determine the number of solutions?
The discriminant is b² - 4ac from the quadratic formula. If positive, the equation has two real solutions. If zero, it has exactly one real solution (a repeated root). If negative, there are no real solutions.
How do you use the discriminant for x² - 6x + 9 = 0?
Calculate b² - 4ac: (-6)² - 4(1)(9) = 36 - 36 = 0. Since the discriminant equals zero, this quadratic has exactly one solution. Factoring confirms: (x-3)² = 0, so x = 3 is the only solution.
Why is checking the discriminant useful before solving a quadratic?
The discriminant instantly reveals whether real solutions exist without going through all the algebra. If the discriminant is negative, no real number satisfies the equation and you can stop — avoiding wasted calculation.