Solving For C: Quadratic Equation With Roots Less Than -2

Hey guys! Today, we're diving into a fun problem involving quadratic equations and their roots. Specifically, we're going to figure out how to find the value of c in a quadratic equation when we know both roots are real and less than -2. Sounds like a challenge? Let’s get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand the problem inside and out. We're given the quadratic equation:

$x^2 - (c+3)x + 9 = 0$

We know two important things about this equation:

1. It has real roots, which we'll call $x_1$ and $x_2$.
2. Both roots are less than -2: $x_1 < -2$ and $x_2 < -2$.

Our mission, should we choose to accept it (and we do!), is to find the value, or rather the range of values, for c that satisfies these conditions. To tackle this, we’ll need to dust off some key concepts about quadratic equations, including the discriminant, the sum and product of roots, and how the graph of a quadratic function behaves. So, buckle up, because we’re about to embark on a mathematical adventure!

The Discriminant: Ensuring Real Roots

First things first, let's address the condition that the roots are real. Remember the discriminant? It's that little expression under the square root in the quadratic formula ($b^2 - 4ac$), and it tells us a lot about the nature of the roots. For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is given by:

$D = b^2 - 4ac$

• If $D > 0$, the equation has two distinct real roots.
• If $D = 0$, the equation has one real root (a repeated root).
• If $D < 0$, the equation has no real roots (two complex roots).

In our case, we need real roots, so the discriminant must be greater than or equal to zero ($D ≥ 0$). Let's calculate the discriminant for our equation:

$x^2 - (c+3)x + 9 = 0$

Here, $a = 1$, $b = -(c+3)$, and $c = 9$. Plugging these values into the discriminant formula, we get:

$D = (-(c+3))^2 - 4(1)(9)$

$D = (c+3)^2 - 36$

For real roots, we need $D ≥ 0$, so:

$(c+3)^2 - 36 ≥ 0$

This inequality is crucial, and we'll come back to it shortly to solve for the possible values of c. It's the first piece of the puzzle in making sure our quadratic equation behaves as expected.

Sum and Product of Roots: Unveiling Root Relationships

Next up, we're going to leverage the relationships between the roots of a quadratic equation and its coefficients. These relationships are super handy for problems like this. For a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots ($x_1$ and $x_2$) are given by:

• Sum of roots: $x_1 + x_2 = -b/a$
• Product of roots: $x_1 * x_2 = c/a$

In our equation, $x^2 - (c+3)x + 9 = 0$, we have:

• $a = 1$
• $b = -(c+3)$
• $c = 9$

So, the sum and product of the roots are:

• $x_1 + x_2 = -(-(c+3))/1 = c+3$
• $x_1 * x_2 = 9/1 = 9$

Now, we know that both roots are less than -2 ($x_1 < -2$ and $x_2 < -2$). This gives us some valuable information about their sum and product. Since both roots are negative, their product is positive (which we already knew since $x_1 * x_2 = 9$). But what about their sum? If both roots are less than -2, their sum must be less than -4:

$x_1 + x_2 < -2 + (-2)$

$x_1 + x_2 < -4$

Substituting the expression for the sum of the roots, we get:

$c + 3 < -4$

This inequality gives us another condition on the value of c, and it's essential for ensuring that both roots are indeed less than -2. Keep this in your mathematical toolkit – it's coming in handy later!

Graphical Interpretation: Visualizing the Roots

Sometimes, the best way to understand a problem is to visualize it. Let's think about what our conditions mean in terms of the graph of the quadratic function $f(x) = x^2 - (c+3)x + 9$. The roots of the equation are the points where the graph intersects the x-axis.

Since both roots ($x_1$ and $x_2$) are less than -2, this means the parabola opens upwards (because the coefficient of $x^2$ is positive) and intersects the x-axis at two points to the left of -2. So, the graph must satisfy these conditions:

1. The parabola opens upwards.
2. It intersects the x-axis at two points.
3. Both intersection points are to the left of -2.

Now, consider the value of the function at $x = -2$, i.e., $f(-2)$. For both roots to be less than -2, the function value at $x = -2$ must be positive. Think about it: if $f(-2)$ were negative, the parabola would have to cross the x-axis somewhere to the right of -2, which contradicts our condition. So, we have:

$f(-2) > 0$

Let's plug $x = -2$ into our function:

$f(-2) = (-2)^2 - (c+3)(-2) + 9$

$f(-2) = 4 + 2(c+3) + 9$

$f(-2) = 4 + 2c + 6 + 9$

$f(-2) = 2c + 19$

So, we need:

$2c + 19 > 0$

This inequality gives us yet another condition on c, making sure the parabola sits in the right position on the graph. It's like we're setting the stage for our roots to make their grand entrance to the left of -2!

Solving the Inequalities

Alright, we've set up three key inequalities based on the conditions of the problem. Now it's time to roll up our sleeves and solve them. This is where the algebra gets real, but don't worry, we'll break it down step by step.

Inequality 1: Discriminant Condition

We found that for real roots, the discriminant must be greater than or equal to zero:

$(c+3)^2 - 36 ≥ 0$

Let's solve this inequality. First, we can rewrite it as:

$(c+3)^2 ≥ 36$

Taking the square root of both sides, we need to consider both the positive and negative roots:

$c+3 ≥ 6$ or $c+3 ≤ -6$

Solving for c in each case:

$c ≥ 3$ or $c ≤ -9$

This tells us that c must be either greater than or equal to 3, or less than or equal to -9 for the quadratic equation to have real roots. We've got our first set of possible values for c – not bad, right?

Inequality 2: Sum of Roots Condition

We also found that the sum of the roots being less than -4 gives us the inequality:

$c + 3 < -4$

Solving for c:

$c < -7$

This inequality adds another layer to our conditions. c must be less than -7 to ensure that the sum of the roots is less than -4, which is essential for both roots to be less than -2. It's like we're narrowing down the possibilities, making our quest for the perfect c even more precise.

Inequality 3: Graphical Condition

Finally, we have the condition from the graphical interpretation:

$2c + 19 > 0$

Solving for c:

$2c > -19$

$c > -19/2$

$c > -9.5$

This inequality tells us that c must be greater than -9.5 to ensure that the parabola is positioned correctly, with both roots to the left of -2. It's like we're fine-tuning the position of the graph to match our requirements.

Combining the Conditions

We've got three inequalities that c must satisfy. Now comes the crucial part: combining them to find the range of values for c that satisfy all conditions simultaneously. This is where we put on our detective hats and piece together the clues.

We have:

1. $c ≥ 3$ or $c ≤ -9$
2. $c < -7$
3. $c > -9.5$

Let's visualize these inequalities on a number line. This will help us see where the conditions overlap.

• Condition 1: $c ≥ 3$ is an interval from 3 to positive infinity, and $c ≤ -9$ is an interval from negative infinity to -9.
• Condition 2: $c < -7$ is an interval from negative infinity to -7.
• Condition 3: $c > -9.5$ is an interval from -9.5 to positive infinity.

Now, we need to find the intersection of these intervals – the region where all three conditions are met. Looking at the number line, we can see that the intersection lies in the interval where:

$-9.5 < c ≤ -9$

This is the final answer! The value of c must be in this range for the quadratic equation to have real roots, both of which are less than -2. It's like we've cracked the code and found the hidden treasure of c values.

Conclusion

Wow, we've really been through it, haven't we? We started with a quadratic equation and a set of conditions, and we navigated through the discriminant, sum and product of roots, graphical interpretation, and a bunch of inequalities. But in the end, we emerged victorious, finding the range of values for c that make everything click.

So, to recap, the value of c for the quadratic equation $x^2 - (c+3)x + 9 = 0$ to have real roots $x_1$ and $x_2$, where $x_1 < -2$ and $x_2 < -2$, is:

$-9.5 < c ≤ -9$

This problem is a fantastic example of how different concepts in algebra come together to solve a single question. We used the discriminant to ensure real roots, the sum and product of roots to relate the roots to the coefficients, and a graphical interpretation to visualize the conditions. It's like we used all the tools in our mathematical toolbox!

Keep practicing problems like this, guys, and you'll become quadratic equation wizards in no time! And remember, math is not just about finding the right answer; it's about the journey, the problem-solving process, and the 'aha!' moments along the way. Keep exploring, keep questioning, and keep having fun with math!