Problem 17 - Algebra 1

Question

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$-3 x^{2}+5 x+5=0$$

Step-by-Step Solution

Answer

The chosen method for solving the given quadratic equation was the quadratic formula. The solutions to the equation are \(x = 5/6 - sqrt(85)/6\) and \(x = 5/6 + sqrt(85)/6\).

Step 1 - Identify coefficients

First identify the coefficients a, b, and c in the quadratic equation. For \(-3x^2 + 5x + 5 = 0\), a = -3, b = 5, and c = 5.

Step 2 - Apply the quadratic formula

Substitute the coefficients a, b, and c into the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / (2a)\). So, the solutions for x in this case are \(x = [-5 ± sqrt((5)^2 - 4*(-3)*(5))] / (2*(-3)) = [-5 ± sqrt(25 + 60)] / -6 = [-5 ± sqrt(85)] / -6.\

Step 3 - Simplify the solutions

Simplify the solutions obtained in the previous step to get the final answers. The two solutions for x are \(x = 5/6 - sqrt(85)/6\) and \(x = 5/6 + sqrt(85)/6\).

Key Concepts

Quadratic Formula

When faced with a quadratic equation such as \( -3x^2 + 5x + 5 = 0 \), the quadratic formula is a powerful tool for finding its solutions. This formula is \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), where \( a \), \( b \), and \( c \) are the coefficients of the equation \( ax^2 + bx + c = 0 \). The symbol \( \pm \) indicates that there will be two solutions, one for each sign.

To make it more accessible, here's a breakdown of using the quadratic formula:

• Identify the coefficients from the quadratic equation.
• Plug these coefficients into the quadratic formula.
• Calculate the discriminant \( b^2 - 4ac \), which is the expression under the square root sign.
• Apply the plus or minus sign to find two possible values for \( x \).
• Simplify your results to get the most reduced form of the solutions.

The effectiveness of the quadratic formula lies in its universality; it can solve any quadratic equation, even when the roots are complex numbers, providing a direct route to the equation's solutions.

Coefficients of Quadratic Equation

In the context of a quadratic equation like \( -3x^2 + 5x + 5 = 0 \), the coefficients are the numbers that multiply the variable \( x \). Specifically, the coefficient \( a \) is the number before \( x^2 \), \( b \) is the number before \( x \) and \( c \) is the constant term with no \( x \). Identifying these coefficients correctly is crucial as they are inputs for the quadratic formula and dictate the curve's shape and position on a graph.

Understanding the role of each coefficient:

• \( a \) determines the curvature and the direction of the parabola. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• \( b \) affects the position of the parabola along the x-axis.
• \( c \) represents the y-intercept, the point where the graph crosses the y-axis.

Therefore, by analyzing the coefficients, one can already gain insights about the quadratic function even before solving the equation.

Simplifying Mathematical Expressions

The process of simplifying mathematical expressions is about making them as straightforward as possible, often to facilitate understanding or to prepare for further operations. Simplifying can include combining like terms, reducing fractions, or, as with the quadratic formula, rationalizing the denominator.

Here's a guide to simplification, specifically after using the quadratic formula:

• Combine any like terms in the numerator and the denominator if possible.
• If there's a square root, consider whether it can be simplified by finding square factors.
• Reduce any common factors between the numerator and the denominator.
• If the denominator is negative, you might want to multiply the top and bottom by -1 for a more standard form, as negative denominators are less common.

Simplification doesn't change the value of the expression; it just presents it in a more digestible form, which is especially helpful for easier comparison or further calculations.