Problem 118 - Intermediate Algebra

Question

Solve each equation. $$x^{2}-\frac{1}{2}(x+1)=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -\frac{1}{2}\).

Step 1: Clear Fraction

To eliminate the fraction, we multiply every term by 2 to get rid of the denominator. The equation becomes: \[ 2x^2 - (x + 1) = 0 \]which simplifies to:\[ 2x^2 - x - 1 = 0 \]

Step 2: Factor the Quadratic Equation

Now, we need to factor the quadratic equation: \[ 2x^2 - x - 1 = 0 \]We look for two numbers whose product is \(-2\times1 = -2\), and sum is \(-1\). The numbers are \(-2\) and \(1\).

Step 3: Rewrite and Factor by Grouping

Rewrite the middle term \(-x\) using the numbers found in Step 2:\[ 2x^2 - 2x + x - 1 = 0 \]Group terms to factor by grouping:\[ (2x^2 - 2x) + (x - 1) = 0 \]Factor out the greatest common factor from each group:\[ 2x(x - 1) + 1(x - 1) = 0 \]

Step 4: Factor Out the Common Binomial

Notice the common binomial \((x - 1)\) in both terms, factor it out:\[ (2x + 1)(x - 1) = 0 \]

Step 5: Solve Each Factor

Set each factor equal to zero and solve:\[2x + 1 = 0 \quad \text{or} \quad x - 1 = 0\]Solving these gives:\[2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}\]\[x = 1\]

Step 6: Conclusion

The solutions to the equation \(x^2 - \frac{1}{2}(x + 1) = 0\) are \(x = 1\) and \(x = -\frac{1}{2}\).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a method used to solve equations of the form \( ax^2 + bx + c = 0 \). The goal is to express the quadratic equation as a product of two binomials. Once the equation is factored, the Zero Product Property is used to find the solutions where each binomial is set equal to zero. Let's break this down with a simple approach you can use every time:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c \).
• Find two numbers that multiply together to give \( ac \) (the product of \( a \) and \( c \)) and add up to \( b \).
• Break apart the middle term \( bx \) using the two numbers found, rewriting the quadratic as a four-term expression.
• Factor by grouping the first two terms and the last two terms. Factor out the common factor in each group if possible.
• If successful, you will have a common binomial factor that will make factoring the entire expression easier.

This systematic approach makes solving quadratics by factoring a clear, step-by-step process.

Solving Quadratic Equations

Solving quadratic equations can be done using several methods, but factoring often offers a straightforward approach when the equation can be factored. After an equation like \( 2x^2 - x - 1 = 0 \) is factored into \((2x + 1)(x - 1) = 0\), each factor gives a separate equation that can be solved individually. Here's what to do:

• Apply the Zero Product Property: If \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). This is the principle that allows you to solve for \( x \) after factoring a quadratic equation.
• Set each factor equal to zero. For example, with \( (2x + 1)(x - 1) = 0 \), set \( 2x + 1 = 0 \) and \( x - 1 = 0 \).
• Solve each equation separately to find the solutions. For \( 2x + 1 = 0 \), solve to get \( x = -\frac{1}{2} \) and for \( x - 1 = 0 \), solve to get \( x = 1 \).

This results in the solutions of the original quadratic equation, and both solutions are valid as long as they satisfy the equation when substituted back in.

Algebraic Fractions

Algebraic fractions appear in equations where one or more terms are fractions. Simplifying these fractions is key to making the rest of the problem easier to handle. In the given exercise, the equation begins with a fractional term: \(x^2 - \frac{1}{2}(x + 1) = 0\). Here's how you can efficiently work with them:

• Find the least common multiple (LCM) of the denominators, if there is more than one fraction, to eliminate the fractions by multiplication.
• Multiply every term in the equation by this LCM. For the exercise, multiplying by 2 removes the fraction, simplifying the equation to \(2x^2 - (x + 1) = 0\).
• This simplification often transforms a more complex equation into a standard quadratic form \( ax^2 + bx + c = 0 \), ready for solving by factoring or other methods.

Once the fraction is cleared, further solving becomes more straightforward. Handling algebraic fractions with confidence is key to efficient problem-solving in algebra.