Problem 77 - College Algebra Essentials

Question

In Exercises \(75-82,\) compute the discriminant. Then determine the number and type of solutions for the given equation. $$ 2 x^{2}-11 x+3=0 $$

Step-by-Step Solution

Answer

The discriminant for the given equation is 97, which is a positive value. Hence, the given quadratic equation will have two distinct real solutions.

Step 1: Identify a, b, and c

From the given quadratic equation \(2x^{2} - 11x + 3 = 0\), the coefficients can be identified as: \(a = 2, b = -11, c = 3\).

Step 2: Compute the discriminant

In a quadratic equation \(ax^{2} + bx + c = 0\), the discriminant \(D\) is calculated using the formula \(D = b^{2} - 4ac\). Substituting the values \(a = 2, b = -11, c = 3\) into the formula, the result will be \(D = (-11)^{2} - 4*2*3 = 121 - 24 = 97\).

Step 3: Determine the number and type of solutions

The nature of the roots of a quadratic equation depends on the value of the discriminant. Since the discriminant \(D = 97\) is positive, the given quadratic equation will have two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental algebraic expression that appears in the form \( ax^2 + bx + c = 0 \). In this formula, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \), as this ensures the equation is indeed quadratic rather than linear. A quadratic equation represents a parabola when graphed on a coordinate plane.
The equation is called "quadratic" because the term "quadratic" is derived from "quad" meaning square, indicating the squared term \( x^2 \). Solving quadratic equations is crucial in many areas of mathematics and science, as they can model various real-world situations like projectile motion, area problems, and economics.
Quadratic equations can be solved using several methods including:

• Factoring, where you express the quadratic equation as a product of its linear factors.
• Completing the square, which transforms the equation into a perfect square.
• Using the quadratic formula, given by \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), which provides an exact solution for any quadratic equation.

Real Solutions

The solutions of a quadratic equation depend on the discriminant, represented by \( D = b^2 - 4ac \). This critical value helps us determine whether the solutions are real or imaginary. The type of solutions that a quadratic equation yields depends heavily on the value of the discriminant.
- If \( D > 0 \), the equation has two distinct real solutions. This is because the square root of a positive number results in two different real numbers.- If \( D = 0 \), the equation has exactly one real solution, often referred to as a repeated or double root. This happens as the square root term in the quadratic formula becomes zero, leading to one solution.- If \( D < 0 \), the equation has no real solutions, but two complex solutions. This is because you cannot take the square root of a negative number in the realm of real numbers, resulting in imaginary results along with real numbers.Understanding these categories of solutions is essential since it dictates how the equation and its graph should be interpreted in real-world problems or statistical data.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are crucial as they represent the points where the graph of the quadratic function crosses the x-axis.
Calculating these roots can be done efficiently with the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). The expression \( \sqrt{b^2-4ac} \) is what we call the discriminant discussed before. Based on its value:

• For \( D > 0 \), the equation has two different x-intercepts.
• For \( D = 0 \), the graph touches the x-axis at one point (the vertex of the parabola).
• For \( D < 0 \), the graph does not intersect the x-axis at any point.

These roots may be referred to as solutions or zeros of the equation, and knowing them is vital in many scientific and engineering calculations where determining precise values or conditions is necessary.