Problem 33 - Algebra 1

Question

Decide how many solutions the equation has. $$8 x^{2}-8 x+2=0$$

Step-by-Step Solution

Answer

The equation \(8x^2-8x+2=0\) has exactly one solution.

Step 1: Identify a, b, and c

From the equation \(8x^2 - 8x + 2 = 0\), we can see that \(a = 8\), \(b = -8\), and \(c = 2\).

Step 2: Calculate the discriminant

The discriminant \(D\) of a quadratic equation can be found using the formula \(D=b^2-4ac\). For our equation, this becomes \(-8^2-4*8*2 = 64-64 = 0\).

Step 3: Determine the number of solutions

Since the discriminant is equal to 0, this means that there is exactly one solution to the equation.Specifically, the quadratic equation \(8x^2-8x+2=0\) has a single real solution.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial of the form |ax^2 + bx + c = 0|, where |a, b, and c| are coefficients with |a ≠ 0|. Quadratic equations are fundamental in algebra and the roots of these equations involve the values of |x| that satisfy the equation.

Though these equations might seem simplistic at first glance, they represent a wide range of phenomena in both mathematics and the real world. For example, they can describe the motion of objects under constant acceleration, the shape of parabolic curves, or even the profit and loss of businesses in certain conditions.

The solution to a quadratic equation can be visualized graphically as the points where the parabola—defined by the equation—intersects with the x-axis. Depending on the specific equation, it might intersect the x-axis at two points, one point, or it might not touch the axis at all.

Solving Quadratic Equations

Solving quadratic equations can be approached in several ways, including factoring, completing the square, graphing, or using the quadratic formula |(-b ± √(b² - 4ac))/(2a)|. Each method has its own set of steps that guide you to finding the values of |x| that make the equation true.

When using the quadratic formula, the expression under the square root, |b² - 4ac|, is known as the discriminant and it plays a pivotal role in determining the nature of the roots. The discriminant tells us whether the roots are real or complex and whether they are distinct or repeated.

• If |D > 0|, the equation has two distinct real roots.
• If |D = 0|, there is one unique real root.
• If |D < 0|, there are two complex conjugate roots.

Understanding the discriminant helps you to anticipate the type and number of solutions without even calculating the roots fully.

Discriminant Formula

The discriminant formula is a powerful tool within the quadratic formula that provides insight into the nature of the solutions of a quadratic equation without actually solving it. It is given by |D = b² - 4ac|. The value of the discriminant, often denoted as |D|, can inform you about the number and type of roots the quadratic equation has.

In the context of our exercise, applying the discriminant formula to the equation |8x^2 - 8x + 2 = 0| means calculating |D = (-8)² - 4(8)(2)|. Simplifying, |D = 64 - 64|, which results in |D = 0|. This indicates that our quadratic equation has a single, repeated real solution, a key piece of information when graphing the function or predicting its behavior.

The accurate calculation of the discriminant is vital for predicting the solution set of a quadratic equation. It's one of the simplest yet most effective ways to quickly assess the nature of the solutions you can expect from a quadratic equation.