Problem 32 - 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 Formula

The equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \). Here, \( a = 2 \), \( b = -8 \), and \( c = 4 \). To find the roots of a quadratic equation, we can use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 2: Calculate the Discriminant

The discriminant is part of the quadratic formula under the square root and is calculated as \( b^2 - 4ac \). Substitute the values:\[(b^2 - 4ac) = (-8)^2 - 4 \times 2 \times 4\]\[ = 64 - 32\]\[ = 32\]

Step 3: Solve Using the Quadratic Formula

Substitute the values for \( a \), \( b \), and the discriminant back into the quadratic formula to find \( x \):\[x = \frac{-(-8) \pm \sqrt{32}}{2 \times 2}\]\[x = \frac{8 \pm \sqrt{32}}{4}\]Since \( \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \), we have:\[x = \frac{8 \pm 4\sqrt{2}}{4}\]

Step 4: Simplify the Solutions

Simplify the expression for each possible value of \( x \):\[x = \frac{8 + 4\sqrt{2}}{4} = 2 + \sqrt{2}\]\[x = \frac{8 - 4\sqrt{2}}{4} = 2 - \sqrt{2}\]So, the solutions are \( x = 2 + \sqrt{2} \) and \( x = 2 - \sqrt{2} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). Quadratic equations often have two solutions, which can be both real or complex numbers. When solving such equations, the quadratic formula allows you to directly calculate the values of \( x \) that make the equation true. The formula is as follows:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are the coefficients from the equation. The symbol \( \pm \) indicates that you usually get two solutions: one by adding the square root and the other by subtracting it.
To apply the formula, you'll need to identify and plug in these coefficients correctly. It's crucial to carefully compute the values step-by-step to ensure accuracy, especially the discriminant under the square root.

Discriminant

The discriminant is the part of the quadratic formula that lies under the square root: \( b^2 - 4ac \). It is a key factor in determining the nature of the roots of a quadratic equation. Calculating the discriminant helps us understand what kind of solutions the quadratic equation will yield. There are primarily three possibilities:

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, the equation has exactly one real solution, also known as a repeated root.
• If the discriminant is negative, the equation has no real solutions, but two complex solutions.

In our example quadratic equation, the discriminant value is 32, which is positive. This indicates that the equation has two real and distinct solutions. It's always beneficial to calculate the discriminant first, even before solving for \( x \), as it guides the next steps in the problem.

Real Solutions

Real solutions are the values of \( x \) that satisfy the quadratic equation and are real numbers, unlike complex solutions involving imaginary numbers. Identifying real solutions is a critical step, especially when dealing with physical or real-world problems. The determination of real solutions involves:

• Calculating the discriminant to verify its sign.
• Using the quadratic formula to systematically solve for the exact values of \( x \).

In the provided problem, the positive discriminant has already told us that real solutions exist. Continuing with the quadratic formula, we substituted the values for \( a \), \( b \), and the discriminant to solve for \( x \), ultimately yielding \( x = 2 + \sqrt{2} \) and \( x = 2 - \sqrt{2} \).
These solutions are not only real but distinct, illustrating how solving quadratic equations can yield specific and meaningful results.