Problem 27 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ 5 a^{2}-2 a-3=0 $$

Step-by-Step Solution

Answer

Question: Using the quadratic formula, solve the equation $$5a^2 - 2a - 3 = 0$$ for a. Answer: The solutions for a are 1 and \(-\frac{3}{5}\).

Step 1: Identify the coefficients a, b, and c

In the given equation, we need to identify the coefficients a, b, and c. The equation is given as: $$ 5a^2 - 2a - 3 = 0 $$ Here, the coefficient a is 5, the coefficient b is -2, and the coefficient c is -3.

Step 2: Apply the quadratic formula

Now we plug in the values for a, b, and c into the quadratic formula: $$ a_1,a_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-3)}}{2(5)} $$

Step 3: Simplify the formula

Next, simplify the quadratic formula: $$ a_1,a_2 = \frac{2 \pm \sqrt{4 + 60}}{10} = \frac{2 \pm \sqrt{64}}{10} $$

Step 4: Calculate the square root term

Now, calculate the square root in the formula: $$ a_1,a_2 = \frac{2 \pm 8}{10} $$

Step 5: Solve for a

Finally, solve for a using the two possible solutions: $$ a_1 = \frac{2 + 8}{10} = \frac{10}{10} = 1 $$ and $$ a_2 = \frac{2 - 8}{10} = \frac{-6}{10} = -\frac{3}{5} $$ Thus, the solutions for a are 1 and \(-\frac{3}{5}\).

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). The solution to this type of equation is found using:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. The term \( \pm \) indicates there are generally two solutions for \( x \), which correspond to potential intersections of the parabola with the x-axis.

This formula is derived from completing the square of the general quadratic equation. The term under the square root, \( b^2 - 4ac \), is called the discriminant, which indicates:

• If it's positive, there are two real solutions.
• If it's zero, there is exactly one real solution.
• If it's negative, the solutions are complex (not real numbers).

This makes the quadratic formula a versatile and powerful method, especially for equations that are difficult to factor.

Solving Equations

Solving quadratic equations involves finding all the possible values of the variable that satisfy the equation. To solve a quadratic equation using the quadratic formula, follow these steps:

• Identify the coefficients: Start by identifying which numbers in your equation correspond to \( a \), \( b \), and \( c \).
• Plug these values into the quadratic formula.
• Simplify inside the square root first (compute the discriminant), then simplify the whole expression.

Step-by-step simplification is crucial. For our example equation, \( 5a^2 - 2a - 3 = 0 \):

• The coefficients are \( a = 5 \), \( b = -2 \), and \( c = -3 \).
• Using the quadratic formula, we substitute \( a \), \( b \), and \( c \) to find the solutions.
• Carefully simplifying ensures accurate solutions and insights into the nature of the roots.

The systematic approach not only solves the problem but enhances understanding of algebraic manipulation.

Coefficients in Algebra

In algebra, coefficients are the numerical factors of terms in equations. Understanding coefficients is key to identifying and working with quadratic equations. For example, in the equation \( 5a^2 - 2a - 3 = 0 \), coefficients are:

• \( a \) in \( 5a^2 \) is 5, representing how much "power" the \( a^2 \) term has.
• \( b \) in \(-2a\) is -2, related to the linear term, affecting the slope or direction the parabola opens.
• \( c \) in \(-3\) is the constant term, which shifts the graph up or down the y-axis.

Recognizing these coefficients allows you to set up your equation for solving with tools like the quadratic formula. By identifying \( a \), \( b \), and \( c \), we can systematically approach finding the solution to our quadratic equation, gaining insights into its graphical representation and the nature of its solutions.