Problem 36 - Algebra 1: Concepts and Skills

Question

Find the value of \(b^{2}\)- 4ac for the equation. $$-8 m^{2}-6 m+3=0$$

Step-by-Step Solution

Answer

The value of the Discriminant (\(b^{2} - 4ac\)) for the given equation is 132.

Step 1: Identify the coefficients a, b, and c

From the equation \(-8 m^{2} - 6 m + 3 = 0\), we have \(a = -8 \), \(b = -6\), and \(c = 3\)

Step 2: Substitute the coefficients into the formula

Substitute \(a = -8\), \(b = -6\), and \(c = 3\) into the formula, \(b^{2} - 4ac\). That gives us \((-6)^{2} - 4*(-8)*3\)

Step 3: Calculate the value

Calculate the value of the expression \((-6)^{2} - 4* (-8)* 3\). This equates to \(36 - (-96) = 132\)

Key Concepts

Coefficients of Quadratic Equations

In any quadratic equation of the form \(ax^2 + bx + c = 0\), the values of \(a\), \(b\), and \(c\) are known as the coefficients. \(a\) is the coefficient of the squared term, \(b\) is the coefficient of the linear term, and \(c\) represents the constant. It's important to correctly identify these coefficients because they are critical for understanding the nature of the quadratic equation and for calculating other aspects of the equation, such as the discriminant.

For example, in the equation \( -8m^2 - 6m + 3 = 0\), \(a = -8\), \(b = -6\), and \(c = 3\). These coefficients dictate the curvature, the direction of the parabola that the equation represents, and they are essential for solving the quadratic equation by various methods.

Calculating Discriminant

The discriminant is a valuable number in the realm of quadratic equations because it reveals the nature of the roots without actually calculating them. It is calculated using the formula \(b^2 - 4ac\), which is derived from the coefficients of a quadratic equation \(ax^2 + bx + c = 0\).

When you evaluate the discriminant, a positive value indicates two distinct real roots, zero means one real root, and a negative value signals that there are no real roots—instead, there are two complex roots. For example, using the coefficients from the equation \( -8m^2 - 6m + 3 = 0\), we calculate the discriminant as \( (-6)^2 - 4(-8)(3)\), which simplifies to \(36 + 96 = 132\). Since 132 is positive, we know the equation has two distinct real roots.

Quadratic Equation

A quadratic equation is an algebraic expression set equal to zero in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are the coefficients, and \(a \eq 0\). The shape of the graph relating to the equation is a parabola, which can either open upwards or downwards depending on the sign of \(a\).

The solutions to a quadratic equation, also known as roots or zeros, can be found using methods such as factoring, completing the square, or applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The quadratic formula method employs the discriminant, which we calculated previously, to determine these root values.

Real-World Applications

• Projectile motion equations in physics often take the form of quadratic equations.
• Economics uses quadratic models to understand cost functions and revenue.
• In engineering, they help design structures with parabolic shapes, such as bridges or satellite dishes.

So, understanding quadratic equations is not just academic—it's essential for practical problem solving in various fields.