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Magazine Chapter 8: Problem 66
Solve. \((1+\sqrt{3}) x^{2}-(3+2 \sqrt{3}) x+3=0\)
Short Answer
Expert verified
The solutions are \( x_1 = 1 \) and \( x_2 = \frac{3 - \sqrt{3}}{2} \).
Step by step solution
01
- Identify coefficients
Identify the coefficients in the quadratic equation. In the equation \( (1+\backslash sqrt{3}) x^{2}-(3+2 \sqrt{3}) x+3=0 \), the coefficients are as follows: \( a = 1 + \sqrt{3} \, b = - (3 + 2 \sqrt{3}) \, c = 3 \).
02
- Use the quadratic formula
The quadratic formula is used to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \). The formula is: \[ x = \frac{-b \pm \backslash sqrt{b^2 - 4ac}}{2a} \]
03
- Substitute coefficients into the formula
Substitute the identified coefficients into the quadratic formula. For our equation, we have: \[ x = \frac{-( -(3 + 2 \sqrt{3})) \pm \backslash sqrt{(3 + 2 \sqrt{3})^2 - 4(1 + \sqrt{3})(3)}}{2(1 + \sqrt{3})} \]
04
- Simplify the equation
Simplify the expression under the square root \( b^2 - 4ac \) and the entire fraction. Calculate \( (3 + 2 \sqrt{3})^2 \) and \(-(4(1 + \sqrt{3})(3)) \): \[ (3 + 2 \sqrt{3})^2 = 9 + 12 \sqrt{3} + 12 = 21 + 12 \sqrt{3} \] \[ 4ac = 4(1 + \sqrt{3})(3) = 12 + 12 \sqrt{3} \] \[ b^2 - 4ac = 21 + 12 \sqrt{3} - (12 + 12 \sqrt{3}) = 9 \]
05
- Solve for x
Now substitute in the simplified values: \[ x = \frac{(3 + 2 \sqrt{3}) \pm 3}{2(1 + \sqrt{3})} \] Solve for both the + and - cases: \[ x_1 = \frac{(3 + 2 \sqrt{3}) + 3}{2(1 + \sqrt{3})} = \frac{6 + 2 \sqrt{3}}{2 + 2 \sqrt{3}} = 1 \] \[ x_2 = \frac{(3 + 2 \sqrt{3}) - 3}{2(1 + \sqrt{3})} = \frac{2 \sqrt{3}}{2 + 2 \sqrt{3}} = \frac{\sqrt{3}}{1 + \sqrt{3}} = \frac{\sqrt{3}(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} = \frac{\sqrt{3} - 3}{-2} = \frac{3-\sqrt{3}}{2} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Quadratic Equations
A quadratic equation is any equation that can be rearranged into the form ax^2 + bx + c = 0, where x represents an unknown variable, and a, b, and c are coefficients with a 鈮� 0. Solving these equations means finding the values of x that satisfy the equation. To solve a quadratic equation, you can use various methods such as factoring, completing the square, and using the quadratic formula.
Here, we'll focus on using the quadratic formula, which offers a straightforward way to find the solutions of any quadratic equation, even when other methods are not applicable.
Here, we'll focus on using the quadratic formula, which offers a straightforward way to find the solutions of any quadratic equation, even when other methods are not applicable.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's a breakdown of each part of the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's a breakdown of each part of the formula:
- \( -b \): This changes the sign of the b coefficient.
- \( \sqrt{b^2 - 4ac} \): This is the discriminant. It shows the number and type of solutions.
- \( 2a \): This is the denominator, which scales the equation appropriately.
Simplification of Radicals
Simplifying radicals is important when solving quadratic equations, especially when using the quadratic formula. A radical in its simplest form has no perfect square factors other than 1 under the radical sign.
- To simplify, factorize the number under the square root into its prime factors, and pair similar factors.
- For example, simplifying \( \sqrt{50} \) gives you \( \sqrt{25 \times 2} = 5\sqrt{2} \)
In our solution, we simplified \( \sqrt{(3 + 2 \sqrt{3})^2 - 4(1 + \sqrt{3})(3)} \) by computing each term separately and reducing it to \( \sqrt{9} = 3 \). Keep practicing these steps to gain confidence.
Intermediate Algebra
Intermediate algebra includes various topics that build a strong foundation for understanding and solving quadratic equations. Key concepts include:
- Understanding how to manipulate and solve polynomial equations.
- Working with radicals and rationalizing denominators.
- \( x_1 = \frac{(3 + 2 \sqrt{3}) + 3}{2(1 + \sqrt{3})} = 1 \)
- \( x_2 = \frac{3 - \sqrt{3}}{2} \)
Mastering techniques like substitution and simplification will help in solving more complex equations. For example, when we solved for x, we needed to combine like terms and rationalize the fraction:
