Problem 36 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 2 x^{2}-8 x+4=0 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Quadratic Equation

The given equation is in the form of a quadratic equation, which can be written as \(ax^2 + bx + c = 0\). In this case, \(a = 2\), \(b = -8\), and \(c = 4\).

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute the values to get \(D = (-8)^2 - 4(2)(4) = 64 - 32 = 32\).

Step 3: Check the Discriminant

Since the discriminant \(D = 32\) is greater than 0, it indicates that the equation has two distinct real solutions.

Step 4: Use the Quadratic Formula

Use the quadratic formula \(x = \frac{{-b \pm \sqrt{D}}}{{2a}}\) to find the solutions. Plug in the values: \(x = \frac{{8 \pm \sqrt{32}}}{{4}}\).

Step 5: Simplify the Square Root

Since \(\sqrt{32} = 4\sqrt{2}\), the solution becomes \(x = \frac{{8 \pm 4\sqrt{2}}}{{4}}\).

Step 6: Simplify the Equation

Simplify the resulting expression: \(x = 2 \pm \sqrt{2}\). Thus, the solutions are \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\).

Key Concepts

real solutions

In the context of quadratic equations, real solutions are the values of the variable that satisfy the equation. These solutions are numbers that do not involve imaginary parts. For a quadratic equation like \( 2x^2 - 8x + 4 = 0 \), the solutions help determine where the quadratic function intersects the x-axis on a graph.

• Real solutions occur when the discriminant of the quadratic equation is greater than or equal to zero.
• If the discriminant is exactly zero, there is exactly one real solution, known as a repeated or double root.
• If the discriminant is greater than zero, the quadratic equation has two distinct real solutions.

In our example, the discriminant is 32, which is greater than zero, indicating two distinct real solutions for the given equation.

discriminant

The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation. It is derived from the equation itself and is given by the formula \( D = b^2 - 4ac \).

• The discriminant is a part of the quadratic equation \( ax^2 + bx + c = 0 \).
• The discriminant tells us about the number and type of solutions or roots a quadratic can have.

In our quadratic equation \( 2x^2 - 8x + 4 = 0 \), plugging the values into the formula gives \( D = (-8)^2 - 4(2)(4) = 32 \). As 32 is greater than zero, this confirms that there are two distinct real solutions.

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has exactly one real solution.
• If \(D < 0\), the equation has no real solutions; instead, it has two complex solutions.

quadratic formula

The quadratic formula is an essential tool for finding solutions to quadratic equations. For any quadratic equation in the form \( ax^2 + bx + c = 0 \), the roots can be found using:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula is derived from the process of completing the square and allows us to compute the solutions directly.

• The "\(\pm\)" symbol means there are two solutions, one with addition and one with subtraction.
• The term under the square root \(\sqrt{b^2 - 4ac}\) is the discriminant, which we discussed earlier.

In our specific problem, by substituting \(a = 2\), \(b = -8\), and \(c = 4\), and knowing the discriminant is 32, we compute:\[x = \frac{8 \pm \sqrt{32}}{4}\] This simplifies to \(x = 2 \pm \sqrt{2}\), giving us the two real solutions: \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\).
Using the quadratic formula is a reliable way to solve any quadratic equation, especially when factoring is challenging or impossible.