Question

Statement \(1:\) For real \(x, \frac{x^{2}+34 x-71}{x^{2}+2 x-7}\) cannot lie in the interval \((5,9)\) Statement \(2:\) For real constant \(\mathrm{a}, \mathrm{b}, \mathrm{c} ; \mathrm{a} \neq 0\) the solutions of equation \(a x^{2}+b x+c=0\) are real if \(b^{2}-4 a c \geq 0\) otherwise imaginary (1) Statement- 1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement- 1 . (2) Statement- 1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1. (3) Statement- 1 is false, Statement-2 is true. (4) Statement- 1 is true, Statement- 2 is false.

Short answer

Option (2) Statement 1 and 2 true but 2 not explaining the first directly correct.

Step-by-step solution

Analyze Statement 1

We need to determine whether the expression \(\frac{x^{2}+34x-71}{x^{2}+2x-7}\) can lie in the interval \((5,9)\). First, understand the behavior of the function by determining the critical points where the numerator or denominator equals zero. These points will help us understand the intervals where the function transforms.

Find the roots of the numerator and denominator

Solve the quadratic equations \(x^2 + 34x - 71 = 0\) and \(x^2 + 2x - 7 = 0\) for their respective roots using the quadratic formula \(-b \pm \sqrt{b^2 - 4ac}/2a\). For the numerator: a = 1, b = 34, c = -71.

Calculate the discriminant of the numerator

\[b^2 - 4ac = 34^2 - 4(1)(-71) = 1156 + 284 = 1440\]. Thus, the roots are real and distinct.

Calculate the roots of the numerator

\[x = \frac{-34 \pm \sqrt{1440}}{2} = \frac{-34 \pm 37.95}{2}\]The roots are approximately \(1.98\) and \(-35.98\).

Calculate the discriminant and roots of the denominator

For \(x^2 + 2x - 7\): Discriminant: \[b^2 - 4ac = 2^2 - 4(1)(-7) = 4 + 28 = 32\]Roots: \[-1 \pm \sqrt{8}\]Which are approximate values \(1.83\) and \(-3.83\)

Understand the behavior near critical points

Analyze the value of the function near all critical points and in between these intervals to determine the range of the function. Check particularly whether values \((5, 9)\) could be achieved.

Validate Statement 1

After testing behavior across all intervals, check: Does the fraction \(\frac{x^2 + 34x - 71}{x^2 + 2x - 7}\) reach values between 5 and 9? The detailed arithmetic reveals no values in that range. Thus, Statement 1 is true.

Analyze Statement 2

Consider whether the statement about properties of quadratic roots \(a x^2 + b x + c\) true independent-check using the discriminant \(b^2 - 4ac \geq 0\) results in real roots if satisfied. Statement 2 is generally true.

Determine linkage of Statement 2 with explanation

Real roots to quadratic descriptor help conclude qualitative behavior but direct bounding interval check involves value range assessment directly utilizing behavior of the specific function.

Final Determination

Knowing that Statement 1 correctness indicated imposes but rooted traits verifying attribute functional range behavior leave judging both statements true; linking sufficient determining behavior range directly; Answer choice correct for (2). Truth separation not intrinsically exclusive mutual direct derivation requirement justified.

Key concepts

Quadratic Equations

Quadratic equations are polynomials of the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a eq 0\). These equations often arise in various real-world scenarios and understanding their properties is essential for solving many algebraic problems. The core of solving quadratic equations lies in finding their roots which are the values of \(x\) that satisfy the equation. This can be done using several methods:

• Factoring: Expressing the quadratic equation as a product of two binomials.
• Completing the Square: Rewriting the equation in a perfect square trinomial form.
• The Quadratic Formula: Using the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] to find the roots directly.

The specific quadratic equations in the original problem are \(x^2 + 34x - 71 = 0\) and \(x^2 + 2x - 7 = 0\). Solving these using the quadratic formula reveals their roots, which helps in understanding the behavior of the rational function in question.

Discriminant

The discriminant of a quadratic equation is a crucial element in determining the nature of its roots. The discriminant is given by the formula: \[ D = b^2 - 4ac \] Depending on the value of the discriminant, we can conclude:

• If \( 0\): The equation has two distinct real roots.
• If \(
• If \( <0\): The equation has two complex (imaginary) roots.

In the original problem, the discriminant helped identify the roots of the quadratics \(x^2 + 34x - 71\) and \(x^2 + 2x - 7\). Calculating the discriminant for each one provided insights into the real and distinct nature of the roots:

• For \(x^2 + 34x - 71\), the discriminant is \(1440\) indicating distinct real roots.
• For \(x^2 + 2x - 7\), the discriminant is \(32\), also indicating distinct real roots.

These roots are essential to analyze the behavior of the rational function \(\frac{x^2 + 34x - 71}{x^2 + 2x - 7}\).

Rational Functions

A rational function is a quotient of two polynomials, written as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). To understand the behavior of a rational function, we need to analyze both the numerator and denominator:

• Critical Points: These are the roots of both the numerator and the denominator. The roots of the denominator are particularly important as they determine the vertical asymptotes (where the function is undefined).
• Behavior at Critical Points: By evaluating the function at points close to the roots, we can determine how it behaves near these critical points.
• Intervals and Sign Analysis: Determining the sign of the rational function in different intervals defined by its critical points helps identify where the function increases or decreases.

In the exercise, the given rational function is \(\frac{x^2 + 34x - 71}{x^2 + 2x - 7}\). Finding the roots of the numerator and denominator allows for an analysis of where the function could potentially fall within the interval \(5, 9\). By evaluating the function at critical points and observing its behavior, it was determined that this interval does not include any values of the function, making the statement regarding its range correct.