Problem 13 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 2 x^{2}+4 x+1=0 $$

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 coefficients

Identify the coefficients of the quadratic equation of the form ax^2 + bx + c = 0. Here, a = 2, b = 4, and c = 1.

Step 2 - Write down the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3 - Substitute the coefficients into the quadratic formula

Substitute a = 2, b = 4, and c = 1 into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{4^2 - 4(2)(1)}}{2(2)} \]

Step 4 - Simplify inside the square root

Simplify the expression inside the square root: \[ x = \frac{-4 \pm \sqrt{16 - 8}}{4} \] which simplifies to \[ x = \frac{-4 \pm \sqrt{8}}{4} \]

Step 5 - Simplify the square root

Simplify the square root of 8: \[ \sqrt{8} = 2\sqrt{2} \], so the equation becomes: \[ x = \frac{-4 \pm 2\sqrt{2}}{4} \]

Step 6 - Simplify the fraction

Simplify each term in the fraction: \[ x = \frac{-4}{4} \pm \frac{2\sqrt{2}}{4} \] which simplifies to: \[ x = -1 \pm \frac{\sqrt{2}}{2} \]

Step 7 - Write the solutions

The solutions to the equation are: \[ x = -1 + \frac{\sqrt{2}}{2} \] and \[ x = -1 - \frac{\sqrt{2}}{2} \]

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is an unknown variable. To solve these equations, we often use the quadratic formula, which is a convenient method for finding the values of \(x\) that satisfy the equation. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula simplifies the process of solving quadratic equations by providing a direct substitution method. We plug in the values of the coefficients \(a\), \(b\), and \(c\), and simplify to find the solutions for \(x\). For instance, if you encounter an equation like \(2x^2 + 4x + 1 = 0\), you would need to identify the coefficients and follow systematic substitution into the formula.

Coefficients

In any quadratic equation of the form \(ax^2 + bx + c = 0\), the symbols \(a\), \(b\), and \(c\) are known as the coefficients. Here’s a simple breakdown:

• The coefficient \(a\) is the number in front of the \(x^2\) term.
• The coefficient \(b\) is the number in front of the \(x\) term.
• The coefficient \(c\) is the constant term, not attached to any variable.

For example, in the equation \(2x^2 + 4x + 1 = 0\), we identify the coefficients as:

• \(a = 2\)
• \(b = 4\)
• \(c = 1\)

Recognizing these coefficients correctly is crucial because they are the inputs you substitute into the quadratic formula. If you mix up the values, you won't get the right answers.

Simplifying Square Roots

When you substitute the coefficients into the quadratic formula, you often find yourself with a square root to simplify. Square roots can sometimes seem tricky, but with the right steps, they can be simplified easily. For example, in our step-by-step solution, we saw the expression:\[ \sqrt{8} = 2\sqrt{2} \] Here is how we simplified it:

• First, recognize that 8 can be factored into 4 * 2.
• Since 4 is a perfect square, take the square root of 4 to get 2 and leave the 2 inside the square root.
• This gives us \(2\sqrt{2}\).

Simplifying square roots will make the overall equation simpler to solve, leading to clearer and more accurate solutions.

Fraction Simplification

The last step often involves simplifying fractions. Simplifying fractions means reducing them to their simplest form. After substituting the coefficients into the quadratic formula and simplifying the square root, you get expressions like: \[ x = \frac{-4 \pm 2\sqrt{2}}{4} \]. To simplify this fraction, you should notice each part is divisible by 4:

• \( \frac{-4}{4} = -1 \)
• \( \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2} \)

Putting it all together, we get the solutions as: \( x = -1 \pm \frac{\sqrt{2}}{2} \). Simplifying fractions helps ensure that your solutions are as clear and as straightforward as possible.