Question

INTERPRETING THE DISCRIMINANT Consider the equation \(\frac{1}{2} x^{2}+\frac{2}{3} x-3=0\) Evaluate the discriminant.

Short answer

The value of the discriminant is approximately 6.44.

Step-by-step solution

Identify coefficients a, b, and c

First identify the coefficients from the quadratic equation \( \frac{1}{2}x^2 + \frac{2}{3}x - 3 = 0 \). Here, a equals \(\frac{1}{2}\), b equals \(\frac{2}{3}\), and c equals -3.

Substitute the coefficients into the discriminant formula

Substitute these coefficient values into the discriminant formula \(b^2 - 4ac\). So it would be \((\frac{2}{3})^2 - 4(\frac{1}{2})(-3) \).

Solve the discriminant formula

Next, solve the expression. It simplifies to \( \frac{4}{9} + 6 \). The sum of these two numbers will be the discriminant.

Final calculation

Add these two numbers to get the final value of the discriminant which is approx 6.44.

Key concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, generally expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not equal to zero since then the equation would be linear, not quadratic.

These equations are fundamental in algebra and pop up in various disciplines, from physics to finance. Understanding how to solve these equations is key to finding solutions that make equations true, known as 'roots' or 'solutions'. There are different methods to solve quadratic equations including factoring, completing the square, using the quadratic formula, and even graphing.

Discriminant Formula

The discriminant is a specific value obtained from a quadratic equation's coefficients, revealing essential properties about its roots. Its formula, denoted \( D \), is \( D = b^2 - 4ac \) where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).

The discriminant can tell us how many real solutions the quadratic equation has; if \( D > 0 \), there are two distinct real solutions. If \( D = 0 \), there is exactly one real solution (a repeated root). And if \( D < 0 \), the equation has no real solutions but two complex ones.

Quadratic Formula

The quadratic formula is a powerful tool to find the solutions of a quadratic equation of the form \( ax^2 + bx + c = 0 \). It states that the roots of the quadratic equation can be found using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where the plus-minus symbol (\(\pm\)) indicates that there are two solutions derived from taking the positive and negative square root of the discriminant.

One of the greatest advantages of the quadratic formula is that it works for any quadratic equation, even when the other methods (like factoring) are difficult or impossible to apply.

Coefficients of a Quadratic Equation

The coefficients of a quadratic equation are the numerical parts of the terms with the variable \( x \). In any quadratic equation of the form \( ax^2 + bx + c = 0 \), \( a \) is the coefficient of the squared term (\( x^2 \)), \( b \) is the coefficient of the first degree term (\( x \)), and \( c \) is the constant term.

Each of these coefficients plays a distinct role in shaping the graph of the quadratic equation, otherwise known as a parabola. The coefficient \( a \) affects the parabola's direction (upwards or downwards) and its width, \( b \) influences the location of the vertex along the x-axis, and \( c \) represents the y-intercept of the parabola.