Lesson 113: Interpreting the Discriminant

New Concept

In the quadratic formula, the expression under the radical sign, $b^2 - 4ac$, is called the discriminant.

What’s next

Next, you'll use this tool to determine the number of real solutions and x-intercepts for several quadratic equations.

Property

In the quadratic formula, the expression under the radical sign, $b^2 - 4ac$, is called the discriminant.

Explanation

Meet the discriminant, the quadratic formula's crystal ball! It's the part under the square root, $b^2 - 4ac$. Before you do all the math, this little number secretly tells you if you'll find two solutions, one, or none at all. It's a super useful shortcut for predicting the outcome of your equation!

Examples

For the equation $x^2 + 5x + 2 = 0$, the discriminant is $(5)^2 - 4(1)(2) = 17$.
For the equation $3x^2 - 2x + 1 = 0$, the discriminant is $(-2)^2 - 4(3)(1) = -8$.
For the equation $x^2 - 6x + 9 = 0$, the discriminant is $(-6)^2 - 4(1)(9) = 0$.

Property

If $b^2 - 4ac > 0$, there are two real solutions. If $b^2 - 4ac = 0$, there is one real solution. If $b^2 - 4ac < 0$, there are no real solutions.

Explanation

The discriminant's sign is like a traffic light for your graph's x-intercepts. Positive means GO, you'll cross the x-axis twice! Zero means SLOW DOWN, you'll touch it just once. Negative means STOP, your parabola completely misses the x-axis. It’s your map to the number of real solutions.

Examples

For $2x^2 - 3x - 5 = 0$, the discriminant is $49$. Since $49 > 0$, there are two real solutions.
For $9x^2 + 6x + 1 = 0$, the discriminant is $0$. There is one real solution.
For $5x^2 - 2x + 3 = 0$, the discriminant is $-56$. Since $-56 < 0$, there are no real solutions.

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:

3. Now, substitute the values for $a$, $b$, and $c$:

4. Simplify the calculation:

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.

Property

If $b^2 - 4ac = 0$, then there is one real solution, which means there is one x-intercept. The solution is called a double root of the equation.

Explanation

When the discriminant is zero, it’s like the two solutions of a quadratic equation have merged into one! The parabola's vertex sits perfectly on the x-axis, giving you one x-intercept. It's called a 'double root' because it technically counts as two identical solutions, making it extra special.

Examples

The equation $x^2 + 8x + 16 = 0$ has a discriminant of $0$, resulting in a double root.
The equation $4x^2 - 12x + 9 = 0$ has a discriminant of $0$, resulting in a double root.
The equation $x^2 - 2x + 1 = 0$ has a discriminant of $0$, resulting in a double root.