Problem 38 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$8 m^{2}+6 m-1=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(m = \frac{-3 + \sqrt{10}}{4}\) and \(m = \frac{-3 - \sqrt{10}}{4}\).

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

In the given quadratic equation \(8m^2 + 6m - 1 = 0\), the coefficient of \(m^2\) (term with power 2) is 'a', the coefficient of 'm' (term with power 1) is 'b' and the constant term is 'c'. So, \(a = 8\), \(b = 6\), and \(c = -1\).

Step 2: Apply the Quadratic Formula

The quadratic formula is \(-b \pm \sqrt{b^2 - 4ac} \div 2a\). Insert the identified values into the formula to get: \(-6 \pm \sqrt{6^2 - 4*8*(-1)} \div 2*8\)

Step 3: Simplify the expressions

Calculate the value inside the square root, simplify the expressions and solve for the two roots 'm'. After simplifying, we get \(m = \frac{-3 \pm \sqrt{10}}{4}\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It typically takes the form \(ax^2 + bx + c = 0\), where \(x\) represents the variable, and \(a\), \(b\), and \(c\) are constants, or coefficients. Quadratic equations always produce a parabola when graphed on a coordinate plane, and they can have zero, one, or two real solutions. This happens because a quadratic equation results from setting a quadratic polynomial to zero.

For instance, in the equation \(8m^2 + 6m - 1 = 0\), the \(m^2\) term ensures that it's a quadratic equation because it has a degree of 2. Understanding the nature of quadratic equations helps us know that the solutions (or roots) can be found where the parabola crosses the x-axis.

• A quadratic equation can have different types of solutions based on its discriminant \(b^2 - 4ac\).
• Graphically, quadratic equations form parabola shapes.
• These equations model a wide range of phenomena, from physics to economics.

Coefficients

In the context of quadratic equations, coefficients are the numbers that multiply the variables of the terms. They are crucial because they define the shape and position of the parabola as well as influence its solutions.

In the equation \(8m^2 + 6m - 1 = 0\), the coefficients are:

• \(a = 8\): This is the coefficient of \(m^2\), and it's the leading coefficient. It affects the width and direction of the parabola. A positive \(a\) opens upwards while a negative one opens downwards.
• \(b = 6\): This coefficient is paired with the linear term \(m\). It influences the parabola's vertex horizontally and vertically.
• \(c = -1\): The constant term, not linked with any variable, affects the height of the parabola along the y-axis.

Identifying these coefficients correctly is essential for applying the quadratic formula accurately. They determine everything from the graph's appearance to the number and nature of solutions.

Solving Quadratic Equations

Solving quadratic equations can be done via several methods, with the quadratic formula being one of the most reliable and general techniques. This formula is derived from completing the square and is applicable when other methods either don't work or are inefficient.

The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula allows us to find the roots of any quadratic equation by substituting the coefficients \(a\), \(b\), and \(c\) into it. The sign \(\pm\) indicates that there are potentially two solutions for each quadratic equation.

• The term \(b^2 - 4ac\) is known as the discriminant. It determines the nature of the roots: real and distinct, real and equal, or complex.
• In the example equation \(8m^2 + 6m - 1 = 0\), substituting \(a = 8\), \(b = 6\), and \(c = -1\) into the formula gives us the roots \(m = \frac{-3 \pm \sqrt{10}}{4}\).

Mastering this formula helps immensely in solving any quadratic equations accurately, making it a vital tool in algebra and beyond.