Problem 73 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. If the solution involves radicals, round to the nearest hundredth. $$x^{2}+3 x-1=0$$

Step-by-Step Solution

Answer

The solutions to the equation are approximately \(x = 0.30\) and \(x = -3.30\).

Step 1: Identify Coefficients

The coefficients for the quadratic equation \(x^{2}+3x-1=0\) are \(a = 1\), \(b = 3\), and \(c = -1\).

Step 2: Use Quadratic Formula

The quadratic formula is given by \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). Substituting \(a = 1\), \(b = 3\), and \(c = -1\) gives \(x = [-3 ± sqrt((3)^2 - 4*1*(-1))] / (2*1)\) which simplifies to \(x = [-3 ± sqrt(9 + 4)] / 2\).

Step 3: Simplify Equation

This step will simplify the quadratic formula to \(x = [-3 ± sqrt(13)] / 2\). When simplifying, make sure to perform operations within brackets or under square roots first.

Step 4: Calculate Roots

Calculate the roots by using the ± symbol in the quadratic formula. So, \(x = ( -3 + sqrt(13)) / 2\) and \(x = ( -3 - sqrt(13)) / 2\)

Step 5: Round to the Nearest Hundredth

The roots calculated will have radicals which need to be rounded to the nearest hundredth, yielding approximately \(x = 0.30\) and \(x = -3.30\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and have various applications in mathematics and science. A quadratic equation is typically expressed in the standard form:

• \( ax^2 + bx + c = 0 \)
• Where \( a, b, \) and \( c \) are constants, and \( a eq 0 \)

This equation format makes them easy to identify. The highest degree term, \( ax^2 \), is what distinguishes quadratic equations from other types.
Quadratic equations can be solved using several methods, including factoring, completing the square, or using the quadratic formula. Each method offers a different approach depending on the nature of the coefficients and the possibility of factoring.
The quadratic formula is particularly useful for finding solutions when the equation does not factor easily.

Solving Equations

Solving quadratic equations can involve a step-by-step process. This involves identifying the type of equation and choosing the most appropriate method to find the solutions.
For the equation \( x^2 + 3x - 1 = 0 \), the quadratic formula is ideal to find the solutions. By substituting the coefficients into the formula:

• \( a = 1 \), \( b = 3 \), \( c = -1 \)
• The quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Using this method helps find the exact values of x, which are also called the roots of the equation.

Radicals

Radicals appear in the solutions of quadratic equations when the expression under the square root sign (the "discriminant") in the quadratic formula is not a perfect square. The discriminant \( b^2 - 4ac \) determines the nature of the roots:

• If it's positive, there are two distinct real solutions.
• If it's zero, there is one real solution.
• If it's negative, the solutions are complex numbers.

In the given equation, solving \( x^2 + 3x - 1 = 0 \) results in radicals because the discriminant \( 13 \) is not a perfect square.
Radicals can sometimes be simplified, but in this case, they lead to approximate values when calculated.

Rounding Numbers

Rounding numbers is an essential skill when working with radicals or decimals, especially in the context of quadratic equations. After solving the equation \( x^2 + 3x - 1 = 0 \) using the quadratic formula, the solutions involve decimals that can be unwieldy. We round these to make our answers more manageable and useful.
Here are handy rounding tips:

• Look at the digit one space to the right of where you want to round.
• If it's 5 or higher, round up the desired decimal place.
• If it's less than 5, keep the desired decimal place as it is.

In this exercise, the radicals were calculated and rounded, yielding approximate values of \( x \approx 0.30 \) and \( x \approx -3.30 \). This rounding is typically done to the nearest hundredth for ease of use and understanding.