Problem 15 - Algebra 1

Question

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$3 x^{2}+17 x+10=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(3x^2 + 17x + 10 = 0\) are \(x = -\frac{2}{3}\) and \(x = -5\). The quadratic formula was used to find these solutions because the equation doesn't lend itself easily to factoring or completing the square.

Step 1: Identify coefficients of quadratic equation

In the quadratic equation \(3x^2 + 17x + 10 = 0\), the coefficients are \(a = 3\), \(b = 17\), and \(c = 10\).

Step 2: Apply the quadratic formula

Use the quadratic formula to solve for \(x\): \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\] Substitute the coefficients into the formula: \[x = \frac{{-17 \pm \sqrt{{(17)^2 - 4*3*10}}}}{{2*3}}\] Simplify to find the solutions.

Step 3: Calculate the solutions

Calculate the discriminant (the part of the formula under the square root): \(b^2 - 4ac = (17)^2 - 4*3*10 = 289 - 120 = 169\). The square root of 169 is 13. Substituting back into the quadratic formula gives the solutions: \[x = \frac{{-17 + 13}}{6} = -\frac{2}{3}\] and \[x = \frac{{-17 - 13}}{6} = -5\]

Key Concepts

Quadratic Formula

One of the most reliable methods for solving quadratic equations is using the quadratic formula. A quadratic equation can be recognized by its standard form, which is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \eq 0 \).

The quadratic formula, \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \], provides a straightforward way to calculate the roots of any quadratic equation. With the quadratic formula, the solution to a quadratic equation is found by substituting the numerical values of the coefficients \( a \) (the coefficient of \( x^2 \)), \( b \) (the coefficient of \( x \)), and \( c \) (the constant term), into the formula.

The beauty of the quadratic formula lies in its universality; it can solve any quadratic equation, whether it has real or complex roots. This formula becomes invaluable when the factors of the quadratic equation are not easily recognizable or when the equation does not factor neatly.

Discriminant of a Quadratic Equation

The discriminant is a key component within the quadratic formula that determines the nature and number of roots of a quadratic equation. It is found within the square root portion of the quadratic formula, \( b^2 - 4ac \), and has significant implications:

• If the discriminant is positive, the equation has two real and distinct solutions.
• If the discriminant is zero, the equation has exactly one real solution, also known as a repeated or double root.
• If the discriminant is negative, the equation has two complex solutions, which are not real numbers.

In the given exercise, the discriminant calculation resulted in \( 169 \), which is a positive number. Thus, we know without completing the entire quadratic formula that there will be two real, distinct solutions. Understanding the discriminant is crucial for students, as it quickly reveals the nature of the roots without the need to solve the entire equation, aiding in the selection of the most efficient solving method.

Coefficients of a Quadratic Equation

The coefficients of a quadratic equation are the numerical factors that precede the variables. In the standard form \( ax^2 + bx + c = 0 \), \( a \) is the coefficient before \( x^2 \) and is crucial because it cannot be zero, as that would make the equation linear rather than quadratic. The coefficient \( b \) accompanies the linear term \( x \) and affects the horizontal alignment and shape of the parabola when graphing the quadratic equation. Lastly, \( c \) is the constant term, which influences the vertical position of the parabola.

Understanding these coefficients is vital for students because they not only dictate the appearance of the graph but also affect the strategy selected for solving the equation and the calculation of the discriminant. In our exercise, we easily identified the coefficients as \( a = 3 \) for the quadratic term and \( b = 17 \) and \( c = 10 \) for the linear and constant terms, respectively. Recognizing and accurately identifying these values is the first step in applying various methods, including the quadratic formula, to find the solutions to the quadratic equation.