Question

Find the real solutions, if any, of each equation. Use any method. $$ \frac{1}{2} x^{2}=\sqrt{2} x+1 $$

Short answer

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

Step-by-step solution

Move all terms to one side

Rewrite the equation by moving all terms to one side to set it equal to zero. Start with: \[ \frac{1}{2} x^2 = \sqrt{2} x + 1 \]Subtract \(\sqrt{2} x + 1\) from both sides to get: \[ \frac{1}{2} x^2 - \sqrt{2} x - 1 = 0 \]

Clear the fraction

Multiply every term by 2 to clear the fraction. This gives: \[ x^2 - 2\sqrt{2} x - 2 = 0 \]

Apply the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( a = 1 \), \( b = -2\sqrt{2} \), and \( c = -2 \). Substitute these values into the formula: \[ x = \frac{-(-2\sqrt{2}) \pm \sqrt{(-2\sqrt{2})^2 - 4(1)(-2)}}{2(1)} \]This simplifies to: \[ x = \frac{2\sqrt{2} \pm \sqrt{8 + 8}}{2} \]\[ x = \frac{2\sqrt{2} \pm 4}{2} \]

Simplify the solutions

Simplify the expression to find the solutions: \[ x = \frac{2\sqrt{2} + 4}{2} \quad \text{or} \quad x = \frac{2\sqrt{2} - 4}{2} \]This simplifies to: \[ x = \sqrt{2} + 2 \quad \text{or} \quad x = \sqrt{2} - 2 \]

Key concepts

quadratic formula

To solve a quadratic equation, we often use the quadratic formula. The quadratic formula provides a systematic approach to finding the solutions of any quadratic equation, which takes the form: ax^2 + bx + c = 0 Here, 'a', 'b', and 'c' are constants. The quadratic formula is expressed as: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Let's break this down:

• The symbol '±' means that there are generally two solutions: one with a plus and one with a minus.
• The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It tells us about the nature of the roots.

This formula is key to finding the values of 'x' that make the equation true. Using it, you can solve any quadratic equation by substituting the corresponding values of 'a', 'b', and 'c'.

simplifying expressions

Simplifying expressions is crucial when working with algebraic equations. This process makes complex equations easier to handle and solve.

Let's consider our equation from the exercise: \( \frac{1}{2} x^2 - \sqrt{2} x - 1 = 0 \) To make computations simpler, we can eliminate the fraction by multiplying every term by 2: \( 2 \times \left( \frac{1}{2} x^2 - \sqrt{2} x - 1 \right) \) This gives us: \( x^2 - 2\sqrt{2} x - 2 = 0 \)

Now, the equation is simpler and does not contain fractions, making it easier to apply the quadratic formula. Simplification helps avoid tricky operations and reduces the likelihood of errors. Always aim to simplify expressions as much as possible.

eliminating fractions

Fractions can complicate solving equations, so it is often helpful to eliminate them early in the process. Here's how we do it:

Given the equation: \( \frac{1}{2} x^2 = \sqrt{2} x + 1 \) First, rewrite it to set one side to zero: \( \frac{1}{2} x^2 - \sqrt{2} x - 1 = 0 \) To eliminate the fraction, multiply every term by the denominator of the fraction, in this case, 2: \( 2 \times \left( \frac{1}{2} x^2 - \sqrt{2} x - 1 \right) = 2 \times 0 \) It simplifies to: \( x^2 - 2\sqrt{2} x - 2 = 0 \)

This step transforms the equation into a simpler form without fractions, making it easier to solve. Eliminating fractions can greatly simplify calculations, especially when using methods like the quadratic formula.