Understanding Quadratic Equation Roots and Properties

The given problem involves a quadratic equation of the form $x^2 + bx + c = 0$, where $\alpha$ and $\beta$ are the roots, with the condition $\alpha < \beta$. We are also given constraints on the coefficients: $b > 0$ and $c < 0$. We need to evaluate the correctness of two statements regarding the relationship between the roots.

Analysing Coefficients and Root Properties

For a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are given by Vieta's formulas:

• Sum of roots: $\alpha + \beta = -\frac{b}{a}$
• Product of roots: $\alpha \beta = \frac{c}{a}$

In our specific equation, $x^2 + bx + c = 0$, we have $a=1$. So the formulas become:

• Sum of roots: $\alpha + \beta = -b$
• Product of roots: $\alpha \beta = c$

Now let's use the given conditions on $b$ and $c$:

• Since $b > 0$, the sum of roots $\alpha + \beta = -b$ must be negative ($\alpha + \beta < 0$).
• Since $c < 0$, the product of roots $\alpha \beta = c$ must be negative ($\alpha \beta < 0$).

Determining the Signs of the Roots

The condition $\alpha \beta < 0$ implies that the roots $\alpha$ and $\beta$ have opposite signs. One root is positive, and the other is negative.

We are given that $\alpha < \beta$. Since they have opposite signs, the smaller root ($\alpha$) must be the negative one, and the larger root ($\beta$) must be the positive one.

So, we conclude that $\alpha < 0$ and $\beta > 0$.

Checking Statement 1: β < - α

The first statement is $\beta < -\alpha$.

From our analysis of the coefficients, we found that the sum of the roots is negative: $\alpha + \beta < 0$.

Let's rearrange this inequality:

$\alpha + \beta < 0$

Subtracting $\alpha$ from both sides gives:

$\beta < -\alpha$

This is exactly the first statement. Since $\alpha + \beta < 0$ is derived directly from the given condition $b > 0$, the statement $\beta < -\alpha$ is always true under the given conditions.

Therefore, statement 1 is correct.

Checking Statement 2: β < |α|

The second statement is $\beta < |\alpha|$.

We determined that $\alpha < 0$. The absolute value of a negative number is its positive counterpart. So, $|\alpha| = -\alpha$ for $\alpha < 0$.

Substituting this into the second statement, the statement becomes:

$\beta < -\alpha$

This is the same inequality as in statement 1. Since we've already shown that $\beta < -\alpha$ is true (because $\alpha + \beta < 0$), the statement $\beta < |\alpha|$ is also true under the given conditions.

Therefore, statement 2 is correct.

Conclusion

Both statements 1 ($\beta < -\alpha$) and 2 ($\beta < |\alpha|$), when evaluated based on the properties derived from the equation's coefficients ($b > 0$ and $c < 0$) and the root ordering ($\alpha < \beta$), are found to be correct.

Let's summarise the key findings:

Checking the statements:

• Statement 1: $\beta < -\alpha$. This is true because $\alpha + \beta < 0$.
• Statement 2: $\beta < |\alpha|$. Since $|\alpha| = -\alpha$ for $\alpha < 0$, this statement is equivalent to $\beta < -\alpha$, which is true.

Thus, both statements are correct.

Revision Table: Quadratic Equation Roots

Additional Information: Quadratic Equation Insights

A quadratic equation $ax^2 + bx + c = 0$ represents a parabola. The roots are the x-intercepts of the parabola. The nature and values of the roots are determined by the coefficients $a$, $b$, and $c$.

• The discriminant, $\Delta = b^2 - 4ac$, tells us about the nature of the roots:
  • If $\Delta > 0$, there are two distinct real roots.
  • If $\Delta = 0$, there is exactly one real root (a repeated root).
  • If $\Delta < 0$, there are two complex conjugate roots.
• In this problem, since $c < 0$ and $a=1$, $4ac = 4c < 0$. Since $b > 0$, $b^2 > 0$. Thus, $\Delta = b^2 - 4c$ will be positive ($b^2 - (negative) > 0$). This confirms that there are two distinct real roots, as stated in the problem.
• The axis of symmetry of the parabola $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. The roots are symmetric about this line. In our case ($a=1$), the axis of symmetry is at $x = -\frac{b}{2}$. Since $b > 0$, the axis of symmetry is at a negative x-value. The roots $\alpha$ and $\beta$ are positioned around this negative value.
• Since $\alpha < \beta$, $\alpha$ is to the left of $-\frac{b}{2}$ and $\beta$ is to the right. The fact that $\alpha < 0$ and $\beta > 0$ means that the parabola crosses the x-axis on both sides of the origin (0,0). Since the product of roots $c$ is the y-intercept when $x=0$, and $c < 0$, the parabola intercepts the y-axis below the x-axis. This visual check aligns with our finding that one root is positive and one is negative.