Question

Solve the equation using the Quadratic Formula. \(2 x^2+\frac{1}{2}=2 x\)

Short answer

\(x = \frac{1}{2}\) is the solution.

Step-by-step solution

Rewrite the Equation in Standard Form

First, rewrite the equation in the standard form of a quadratic equation \(ax^2 + bx + c = 0\). It might require some calculations to move terms around. From \(2x^2 + \frac{1}{2} = 2x\), subtract 2x from both sides to get \(2x^2 - 2x + \frac{1}{2} = 0\).

Identify the Values of a, b, and c

Next, identify the coefficient of \(x^2\) as a, the coefficient of x as b, and the constant term as c. Here, a = 2, b = -2, and c = \(\frac{1}{2}\). These values are used in the quadratic formula.

Calculate the Discriminant

The discriminant is used to determine how many real solutions the equation has. The discriminant is calculated by \((b^2 - 4ac)\). Hence, substitute \(a = 2\), \(b = -2\), \(c = \frac{1}{2}\) into the formula. The discriminant will be \( -2^2 - 4*2*\frac{1}{2} = 4 - 4 = 0\). The equation has one real solution, since the discriminant equals zero (b^2 - 4ac = 0).

Solve the Equation with the Quadratic Formula

Since the equation has one real solution, use the quadratic formula to find it. Substitute a, b, and c into the formula, the solution will be \(x = \frac{-(-2) ± \sqrt{(-2)^2 - 4*2*\frac{1}{2}}}{2*2}\). Simplify to get \(x = \frac{2 ± \sqrt{0}}{4}\). Therefore, \(x = \frac{1}{2}\).

Key concepts

Quadratic Equation

Quadratic equations are polynomial equations of the second degree. They have the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. Solving a quadratic equation means finding the values of \( x \) that make the equation true. These values are often referred to as the "roots" or "solutions."

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages, and the quadratic formula is particularly useful when other methods are complex or impossible to apply.

• It handles any quadratic equation.
• It's a formulaic approach: plug in numbers and solve.

Discriminant

The discriminant is a crucial part of solving quadratic equations. It is found within the quadratic formula under the square root: \( b^2 - 4ac \). The value of the discriminant can tell us a lot about the nature of the solutions.

Here’s how the discriminant influences possible solutions:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real solutions.
• If \( b^2 - 4ac = 0 \), the equation has exactly one real solution, as seen in this exercise.
• If \( b^2 - 4ac < 0 \), the equation has no real solutions, but two complex solutions.

Understanding the discriminant is key because it tells us whether we'll have real solutions to find and how many there will be.

Real Solutions

Real solutions for a quadratic equation refer to the values of \( x \) that satisfy the equation and are real numbers. Unlike complex solutions, which have imaginary parts, real solutions are numbers you would typically find on the number line.

The discriminant helps determine if real solutions exist:

• Positive discriminant: two real solutions.
• Zero discriminant: one real solution. This was the case in the given equation, resulting in \( x = \frac{1}{2} \).
• Negative discriminant: no real solutions.

In practical applications, finding real solutions is often the goal, as they correspond to physically meaningful answers in contexts like physics and engineering.

Standard Form

A quadratic equation's standard form is \( ax^2 + bx + c = 0 \). Rewriting equations in this form is essential because it allows us to easily use the quadratic formula or other methods to solve the equation.

Here's the process to convert an equation into standard form:

• Ensure all the terms are on one side of the equation, leaving zero on the other side.
• Arrange terms in descending order of the power of \( x \): \( x^2 \), \( x \), and then constant terms.

In our exercise, we transformed \( 2x^2 + \frac{1}{2} = 2x \) into \( 2x^2 - 2x + \frac{1}{2} = 0 \) by moving all terms to one side. This enables us to apply the quadratic formula accurately.