Problem 90 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 2 x^{2}+3 x=1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -3/4 + sqrt(17)/4\) and \(x = -3/4 - sqrt(17)/4\).

Step 1: Simplify to standard form

Rewrite the equation in standard form. In this case subtract 1 on both sides to have the equation \(2x^2 + 3x - 1 = 0\)

Step 2: Identify a, b, and c

From the simplified equation, identify the coefficients a, b, and c. In this case, a=2, b=3, c=-1.

Step 3: Apply the quadratic formula

The quadratic formula is \(x = [-b ± sqrt(b^2 - 4ac) ] / 2a\). Apply this formula with a=2, b=3, and c=-1, we get \(x = [-3 ± sqrt((3)^2 - 4*2*(-1) ) ] / (2*2)\)

Step 4: Simplify the solution

Simplify to find the solution: \(x = [-3 ± sqrt(9+8)] / 4 = [-3 ± sqrt(17)] / 4\). So the solutions are \(x = -3/4 + sqrt(17)/4\) and \(x = -3/4 - sqrt(17)/4\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It allows us to find the solutions, or roots, of any quadratic equation in the standard form. The formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula can be used when the quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here’s how it works:

• \(-b\): Take the opposite of the coefficient of \(x\).
• \(b^2 - 4ac\): Known as the discriminant, this part helps determine the nature of the roots.
• \(\pm\): Indicates there are usually two solutions, corresponding to plus and minus.
• \(\sqrt{}\): Calculate the square root of the discriminant to proceed.
• \(\frac{\underline{\phantom{xx}}}{2a}\): Divide the entire expression by two times the coefficient of \(x^2\).

Using the quadratic formula can seem tricky at first, but with practice, it becomes straightforward.

Standard Form of Quadratic Equation

Understanding the standard form of a quadratic equation is crucial for utilizing the quadratic formula. The standard form is represented as:
\( ax^2 + bx + c = 0 \)
where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Here’s why it matters:

• Consistency: Standardizing the equation helps in applying various solving methods like factoring and the quadratic formula.
• Identifying Coefficients: Recognizing \(a\), \(b\), and \(c\) is essential for plugging them into the quadratic formula.
• Comparing Terms: Aligning the equation allows for easy identification of quadratic, linear, and constant terms.

In our example, rewriting \(2x^2 + 3x = 1\) as \(2x^2 + 3x - 1 = 0\) aligns it to standard form, making it ready for further solutions.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. Several methods can be used:

• Factoring: Break down the quadratic equation into factors, if possible, to find the solutions.
• Completing the Square: Transform the equation into a perfect square trinomial to solve for \(x\).
• Quadratic Formula: Directly substitute the coefficients \(a\), \(b\), and \(c\) into the formula to find solutions.

The quadratic formula is particularly useful when the equation is complex or doesn’t factor easily. Take the example \(2x^2 + 3x - 1 = 0\). By using the quadratic formula, with \( a = 2 \), \( b = 3 \), and \( c = -1 \), we obtain two solutions:
\[ x = \frac{-3 \pm \sqrt{17}}{4} \]
These solutions provide the points where the quadratic equation reaches zero on the coordinate plane. Thus, solving quadratic equations opens up the way to understanding their graphs and behavior.