Problem 54 - Algebra 2

Question

If \(2 x^{2}-5 x-9=0,\) then \(x\) could be approximately equal to which of the following? A. \(-1.12\) B. 1.54 C. 2.63 D. 3.71

Step-by-Step Solution

Answer

x could be approximately equal to 3.71 (option D).

Step 1: Understand the Equation

The equation given is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -5 \), and \( c = -9 \). We need to solve for \( x \) to find the approximate value.

Step 2: Use the Quadratic Formula

To find the roots of the quadratic equation, we apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = -5 \), and \( c = -9 \).

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is given by \( b^2 - 4ac \). Calculate it: \[ \Delta = (-5)^2 - 4(2)(-9) \] \[ \Delta = 25 + 72 = 97 \].

Step 4: Calculate the Roots

Using the discriminant \( \Delta = 97 \) in the quadratic formula: \[ x = \frac{5 \pm \sqrt{97}}{4} \]. The two possible solutions are: \[ x_1 = \frac{5 + \sqrt{97}}{4} \] and \[ x_2 = \frac{5 - \sqrt{97}}{4} \].

Step 5: Approximate the Roots

Calculate the approximate values using a calculator: \( \sqrt{97} \approx 9.85 \). This gives us: \[ x_1 = \frac{5 + 9.85}{4} = \frac{14.85}{4} \approx 3.71 \] and \[ x_2 = \frac{5 - 9.85}{4} = \frac{-4.85}{4} \approx -1.21 \].

Step 6: Match with Given Options

The approximate solutions are \( x_1 \approx 3.71 \) and \( x_2 \approx -1.21 \). Match these with the given options. The closest matches are \( 3.71 \) and there's no option for \( -1.21 \), but \( -1.12 \) is close.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. If you have an equation in the format of \( ax^2 + bx + c = 0 \), the quadratic formula helps find the values of \( x \) that solve the equation. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are the coefficients of the equation. Following the formula, you start by calculating the discriminant \( b^2 - 4ac \). You take the square root of the discriminant and use it to find two possible values for \( x \), which are the solutions to the quadractic equation. This formula is incredibly powerful because it works for any quadratic equation, guaranteeing that you can find the roots.

• Step 1: Identify \( a \), \( b \), and \( c \) in the equation.
• Step 2: Use the quadratic formula to calculate the possible solutions.

Discriminant

The discriminant is a key part of the quadratic formula that affects the nature of the roots of a quadratic equation. It's the term under the square root in the quadratic formula, \( b^2 - 4ac \), and it's often denoted as \( \Delta \). By calculating the discriminant, you can tell important things about the roots:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (or a double root).
• If \( \Delta < 0 \), there are no real roots, but two complex roots.

In our example, the discriminant is calculated as \( (-5)^2 - 4 \times 2 \times (-9) = 25 + 72 = 97 \). Because \( 97 \) is greater than zero, we know that the equation has two distinct real roots. This knowledge is crucial because it indicates whether calculating the roots further will yield real or complex numbers.

Approximating Roots

Once the discriminant is calculated and integrated into the quadratic formula, the next step is to find the approximations of the roots. This involves calculating the values, often using a calculator for precision, since square roots of discriminants like \( 97 \) may not perfectly resolve without computing tools.Here is how you approach finding approximate roots:

• Calculate \( \sqrt{97} \approx 9.85 \), using your calculator.
• Plug this approximation into the quadratic formula:
• \( x_1 = \frac{5 + 9.85}{4} = 3.71 \)
• \( x_2 = \frac{5 - 9.85}{4} = -1.21 \)

Approximating roots is a practical step where you transition from the algebraic exactness to a real-world usable number, especially useful when exact roots are irrational as seen here. This approximate calculation helps in matching the results with answer choices in multiple-choice scenarios.