Problem 17 - Algebra 1: Concepts and Skills

Question

Find the discriminant of the quadratic equation. \(2 x^{2}+8 x=-8\)

Step-by-Step Solution

Answer

The discriminant of the equation \(2x^{2} + 8x + 8 = 0\) is 0

Step 1: Rearrange the equation

To get the equation into the form \(ax^{2} + bx + c = 0\), move -8 from the right side to the left side of the equation. This gives the equation \(2x^{2} + 8x + 8 = 0\)

Step 2: Identify the coefficients

In the standard form equation \(ax^{2} + bx + c = 0\), a is the coefficient of \(x^{2}\), b is the coefficient of x, and c is the constant. Therefore, for the equation \(2x^{2} + 8x + 8 = 0\), \(a = 2\), \(b = 8\) and \(c = 8\)

Step 3: Calculate the discriminant

Now plug these values into the discriminant formula \(b^{2} - 4ac\), giving \(8^{2} - 4*2*8\). This simplifies to \(64 - 64 = 0\)

Key Concepts

Discriminant Formula

The discriminant is a powerful tool used in algebra to determine the nature of the roots of a quadratic equation without actually solving it. To find the discriminant of a quadratic equation in the form of \(ax^{2} + bx + c = 0\), we use the discriminant formula, which is \(b^{2} - 4ac\).

The discriminant can tell us whether the quadratic equation has:

• Two distinct real roots (if discriminant > 0)
• One real root (if discriminant is 0)
• No real roots, but two complex roots (if discriminant < 0)

In the exercise given, the discriminant calculation for the equation \(2x^{2} + 8x + 8 = 0\) results in a value of 0, suggesting that this specific quadratic equation has one real root, or in other words, it has two identical real roots.

Quadratic Equations

Quadratic equations are a type of polynomial equation of second degree, meaning they have the highest exponent of 2 for the variable x. The standard form of a quadratic equation is \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) must be non-zero.

These equations graphically represent parabolas when plotted on a coordinate plane, opening either upwards or downwards, depending on the sign of \(a\). Understanding the structure of quadratic equations is essential, as they appear frequently in mathematical problems and real-world applications alike.

Solving Quadratics

Solving quadratic equations involves finding the value(s) of \(x\) that satisfy the equation. There are multiple methods to tackle these equations:

• Factoring the quadratic if it can easily be broken down into binomials.
• Using the quadratic formula \(x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}}\) which works for all types of quadratic equations.
• Completing the square, which involves manipulating the equation to form a perfect square trinomial.
• Graphing the quadratic equation and finding the x-intercepts of the parabola.

For all these methods, the discriminant plays a crucial role because it indicates the nature and the number of solutions before one even starts solving the equation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, division, etc.). They're fundamental in forming equations and can vary greatly in complexity from simple such as \(2x + 3\), to more complex ones involving exponents like \(2x^{2} + 8x + 8\).

Understanding these expressions is the cornerstone of algebra. We manipulate these expressions to solve for unknown variables, simplify the expressions themselves, or even substitute values to find numeric results. Reading and writing algebraic expressions is a skill that's honed over time and essential for solving quadratic equations and other algebraic problems.