Problem 10 - Algebra 1

Question

Using THE DISCRIMINANT Tell if the equation has two solutions, one solution, or no real solution. $$2 x^{2}-4 x+3=0$$

Step-by-Step Solution

Answer

The equation \(2x^2 - 4x + 3 = 0\) has no real solutions.

Step 1: Identify coefficients

From the equation \(2x^2 - 4x + 3 = 0\), we identify the coefficients as \(a=2\), \(b=-4\), and \(c=3\).

Step 2: Find the discriminant

We calculate the discriminant using the formula \(b^{2}-4ac\), which becomes \((-4)^{2} - 4*2*3\).

Step 3: Compute the discriminant

Calculate the expression \((-4)^{2} - 4*2*3 = 16 - 24 = -8\).

Step 4: Determine the number of solutions

The number of solutions is determined by the sign of the discriminant. If the discriminant is greater than 0, there are two distinct solutions. If the discriminant is equal to 0, there is one solution. If the discriminant is less than 0, there are no real solutions. In this case, the discriminant is -8, which is less than 0.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation that involves the square of a variable. A typical quadratic equation has the form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). Quadratic equations are characterized by the highest degree term being \( x^2 \). This form is common in mathematics because it appears in various real-world scenarios such as physics, engineering, and finance.
It's important to notice that these equations are called "quadratics" because the word "quadratic" comes from "quad," meaning square. The key component that makes them different from linear equations is the \( x^2 \) term. Solving these equations aims to find the value(s) of \( x \) that make the equation true.
When dealing with quadratic equations, we often use different methods like factoring, completing the square, or using the quadratic formula. Each method offers unique benefits depending on the specific equation.

Coefficients

In the context of the quadratic equation \( ax^2 + bx + c = 0 \), coefficients are the numerical constants \( a \), \( b \), and \( c \) in the equation. Each coefficient serves a specific role:

• \( a \) is the coefficient of \( x^2 \); it determines the parabola's "direction" (upward if positive, downward if negative) and "width" (wider if \( |a| \) is smaller, narrower if \( |a| \) is larger).
• \( b \) is the coefficient of \( x \); it influences the line's slope and affects the position of the parabola along the x-axis.
• \( c \) is the constant term; it shifts the parabola up or down along the y-axis without affecting its shape.

Since these coefficients affect the shape and the position of the parabola graph, they are also crucial in determining the solutions or roots of the equation. Recognizing and correctly using these coefficients in calculations, such as finding the discriminant, is key to solving quadratic equations effectively.

Real Solutions

Real solutions of quadratic equations refer to the actual values of \( x \) that satisfy the equation when plotted on a real number line. The number of real solutions depends mainly on the discriminant, \( b^2 - 4ac \), a special part of the quadratic formula. Here's how it works:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), the equation has two distinct real solutions. This means the parabola intersects the x-axis at two points.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), there is exactly one real solution. The vertex of the parabola touches the x-axis, indicating one repeated root.
• If the discriminant is negative, \( b^2 - 4ac < 0 \), there are no "real" solutions. Instead, the solutions are complex or imaginary, as the parabola does not intersect the real number line.

Understanding the discriminant helps in predicting how many solutions an equation has, and whether those solutions are real numbers. This insight is vital for interpreting mathematical models and equations in the real world.