Problem 5 - Algebra 2

Question

Find the exact solutions by using the Quadratic Formula. \(2 x^{2}-4 x+1=0\)

Step-by-Step Solution

Answer

The solutions are \( x = 1 + \frac{\sqrt{2}}{2} \) and \( x = 1 - \frac{\sqrt{2}}{2} \).

Step 1: Identify the Quadratic Formula

The quadratic formula is used to find solutions for equations of the form \( ax^2 + bx + c = 0 \). It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this equation, \(a = 2\), \(b = -4\), and \(c = 1\).

Step 2: Compute the Discriminant

The discriminant of a quadratic equation is the expression inside the square root of the quadratic formula: \(b^2 - 4ac\). Compute it for our equation:\[(-4)^2 - 4 \times 2 \times 1 = 16 - 8 = 8\].

Step 3: Apply the Quadratic Formula

Now that we have \(b^2 - 4ac = 8\), plug in the values of \(a\), \(b\), and the discriminant into the quadratic formula:\[ x = \frac{-(-4) \pm \sqrt{8}}{2 \times 2} = \frac{4 \pm \sqrt{8}}{4} \].

Step 4: Simplify the Square Root

Simplify \(\sqrt{8}\) by expressing it in simplest radical form: \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \].

Step 5: Simplify the Expression Further

Substitute \(2\sqrt{2}\) into the formula, and then simplify: \[ x = \frac{4 \pm 2\sqrt{2}}{4} \]. This results in two possible solutions: \[ x = \frac{4 + 2\sqrt{2}}{4} = 1 + \frac{\sqrt{2}}{2} \] and \[ x = \frac{4 - 2\sqrt{2}}{4} = 1 - \frac{\sqrt{2}}{2} \].

Key Concepts

Discriminant

The concept of the discriminant is a crucial aspect of using the quadratic formula. It helps us determine what type of solutions we can expect from a quadratic equation of the form \(ax^2 + bx + c = 0\). The discriminant is the part of the formula that is inside the square root, calculated as \(b^2 - 4ac\). This value plays a significant role in identifying the nature of the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, meaning the roots are repeated.
• If the discriminant is negative, the equation has no real roots, but instead two complex roots.

For our specific problem \(2x^{2} - 4x + 1 = 0\), we computed the discriminant to be 8, which is positive. This tells us that there are two distinct real solutions to the equation. Calculating the discriminant helps simplify the process of solving quadratic equations, as it provides insight into the nature of the solutions without fully solving the equation at first.

Radical simplification

Radical simplification is an essential skill when dealing with quadratic equations, particularly within the quadratic formula. Simplifying radicals often makes equations easier to solve and understand. In mathematics, simplifying a radical typically involves expressing a square root in its simplest form by factoring out perfect squares.

For the equation \(2x^2 - 4x + 1 = 0\), we needed to simplify \(\sqrt{8}\) when applying the quadratic formula. To simplify \(\sqrt{8}\), we follow these steps:

• First, factor 8 into \(4 \times 2\).
• Recognize that 4 is a perfect square, so \(\sqrt{4} = 2\).
• Express \(\sqrt{8}\) as \(\sqrt{4 \times 2} = 2\sqrt{2}\).

This simplification helps us easily substitute the simpler radical expression back into the formula, allowing the quadratic solutions to be expressed more clearly. Radical simplification is crucial as it reduces confusion and ensures solutions are presented in the cleanest and most interpretable manner possible.

Equation solving

Solving a quadratic equation using the quadratic formula involves substituting the given coefficients and the discriminant into the formula to find the roots. For equations like \(2x^2 - 4x + 1 = 0\), once we have calculated the discriminant and simplified any radicals, the process of solving becomes relatively straightforward.

To solve equation \(2x^2 - 4x + 1 = 0\) using the quadratic formula, perform the following steps:

• Identify the values of \(a\), \(b\), and \(c\) from the equation.
• Plug these values into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Use the previously simplified radical form to express the solutions cleanly.
• Finally, calculate the two solutions by considering both the '+' and '−' scenarios in the formula.

For this example, inserting the simplified \(\sqrt{8}\) as \(2\sqrt{2}\), we reach the solutions \(x = 1 + \frac{\sqrt{2}}{2}\) and \(x = 1 - \frac{\sqrt{2}}{2}\). Solving quadratic equations in this manner provides an exact result, avoiding any approximation or rounding errors, and ensures the solutions are expressed in the most precise form. This method is especially useful in a variety of mathematical and real-world applications.