Understanding quadratic inequalities can be easier if you use test points. This method helps you find the solution sets, which show where the quadratic function is above or below the x-axis. Quadratic inequalities usually look like this:

$ax^2 + bx + c < 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are numbers (we call them constants).

Here’s a simple step-by-step guide to solve them.

Step 1: Solve the Quadratic Equation

First, change the inequality into an equation. For example, if you have:

$x^2 - 4x + 3 < 0$,

start by solving the equation:

$x^2 - 4x + 3 = 0$.

You can either factor it or use the quadratic formula. In this case, factoring works well:

$(x - 1)(x - 3) = 0$.

This gives us two solutions: $x = 1$ and $x = 3$. These are the points where our quadratic touches the x-axis.

Step 2: Determine Intervals

Next, we divide the number line into parts using the roots we found. For our quadratic $x^2 - 4x + 3$, we have these intervals:

1. $(-\infty, 1)$
2. $(1, 3)$
3. $(3, \infty)$

Step 3: Choose Test Points

Now, we pick some test points from each interval to see where the inequality is true.

• For the interval $(-\infty, 1)$, choose $x = 0$.
• For the interval $(1, 3)$, try $x = 2$.
• For the interval $(3, \infty)$, go with $x = 4$.

Step 4: Substitute Test Points into the Inequality

Now, plug these test points into the original inequality $x^2 - 4x + 3 < 0$.

• For $x = 0$: $0^2 - 4(0) + 3 = 3 \quad (\text{not } < 0)$

• For $x = 2$: $2^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \quad (< 0)$

• For $x = 4$: $4^2 - 4(4) + 3 = 16 - 16 + 3 = 3 \quad (\text{not } < 0)$

Step 5: Analyze the Results

Now, let’s see what we found:

• In the interval $(-\infty, 1)$, the result is positive, so it doesn’t satisfy the inequality.
• In the interval $(1, 3)$, the result is negative, which means it satisfies the inequality.
• In the interval $(3, \infty)$, the result is also positive, so this one doesn’t satisfy the inequality either.

Final Conclusion

So, the solution for the inequality $x^2 - 4x + 3 < 0$ is just the interval $x \in (1, 3)$.

Using test points is a helpful way to see where the quadratic function is positioned in relation to the x-axis. It’s a great strategy for finding solution sets.

Next time you face a quadratic inequality, remember this method! It will help make the problem easier to handle and a lot less scary!