Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by completing the square. Show each step. $$ 2 x^{2}-3 x-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3 + \sqrt{17}}{4} \) and \( x = \frac{3 - \sqrt{17}}{4} \).

Step 1: Move Constant to the Right Side

Start by moving the constant term to the right side of the equation: \[ 2x^2 - 3x = 1 \]

Step 2: Divide by Leading Coefficient

Divide every term by the coefficient of \(x^2\) to make it 1:\[ x^2 - \frac{3}{2}x = \frac{1}{2} \]

Step 3: Complete the Square

To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is \(-\frac{3}{2}\), so half of it is \(-\frac{3}{4}\), and its square is \(\left(-\frac{3}{4}\right)^2 = \frac{9}{16}\). Add \(\frac{9}{16}\) to both sides:\[ x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{1}{2} + \frac{9}{16} \]

Step 4: Simplify Both Sides

Rewrite the left side as a perfect square and simplify the right side:\[ (x - \frac{3}{4})^2 = \frac{8}{16} + \frac{9}{16} = \frac{17}{16} \]

Step 5: Solve for x by Taking Square Roots

Take the square root of both sides to solve for \(x\):\[ x - \frac{3}{4} = \pm \sqrt{\frac{17}{16}} \]\[ x - \frac{3}{4} = \pm \frac{\sqrt{17}}{4} \]

Step 6: Solve for x

Add \(\frac{3}{4}\) to both sides to isolate \(x\):\[ x = \frac{3}{4} \pm \frac{\sqrt{17}}{4} \]This gives us two solutions: \[ x = \frac{3 + \sqrt{17}}{4} \] and \[ x = \frac{3 - \sqrt{17}}{4} \]

Key Concepts

Completing the Square

Completing the square is a method used in algebra to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The idea is to transform this equation into a perfect square trinomial, something like \((x - p)^2 = q\), which is easier to solve.

• Start by moving the constant term to the other side of the equation.
• If the coefficient of \(x^2\) is not 1, divide the entire equation by this coefficient to simplify.
• Take half the coefficient of \(x\), square it, and add it to both sides of the equation.
• This creates a perfect square trinomial on the left-hand side.

By following these steps, the quadratic equation becomes more manageable, leading to solutions via taking the square root of both sides.

Quadratic Formula

The quadratic formula is another powerful tool for solving quadratic equations. The formula is derived from the method of completing the square and is used to find the roots of any quadratic equation \(ax^2 + bx + c = 0\). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides solutions for \(x\) in terms of \(a\), \(b\), and \(c\), the coefficients of the quadratic equation.

• Calculate the discriminant, \(b^2 - 4ac\), which determines the nature of the roots.
• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root.
• If the discriminant is negative, there are two complex roots.

This formula is especially useful because it can solve any quadratic equation without needing to manipulate the equation extensively.

Algebraic Techniques

Algebraic techniques involve various strategies and operations for rearranging and simplifying mathematical equations or expressions. These techniques are essential in solving quadratic equations and include:

• Transposing terms to move them from one side of the equation to the other.
• Combining like terms to simplify expressions.
• Factoring equations to make them easier to solve.
• Applying operations such as addition, subtraction, multiplication, and division on both sides of the equation to maintain balance.

Mastering these techniques helps in solving complex algebraic problems by breaking them down into simpler, more manageable parts. Understanding and practicing these methods allow students to confidently navigate through mathematical challenges.

Polynomials

Polynomials are mathematical expressions made up of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A quadratic polynomial is a special kind of polynomial that includes a term with a variable raised to the power of two, usually in the form \(ax^2 + bx + c\).

• The degree of a polynomial is determined by the highest exponent of its variable.
• Quadratic polynomials have a degree of two, which signifies the highest power of the variable is two.
• They can be classified based on the number of terms, such as monomials, binomials, or trinomials.

Understanding polynomials is fundamental in algebra, as they form the basis for more complex equations and are pivotal in mathematical models across various applications.