Problem 122 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(25 q^{2}+30 q+9=0\)

Step-by-Step Solution

Answer

The solution is \( q = -\frac{3}{5} \).

Step 1: Identify coefficients

Identify the coefficients in the quadratic equation. Here, the equation is in the form \[25q^2 + 30q + 9 = 0\] where \(a = 25\), \(b = 30\), and \(c = 9\).

Step 2: Write down the Quadratic Formula

The Quadratic Formula to solve any equation of the form \(ax^2 + bx + c = 0\) is:\[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Substitute the coefficients into the formula

Substitute \(a = 25\), \(b = 30\), and \(c = 9\) into the Quadratic Formula:\[ q = \frac{-30 \pm \sqrt{30^2 - 4 \cdot 25 \cdot 9}}{2 \cdot 25} \]

Step 4: Simplify inside the square root

Simplify the expression under the square root (the discriminant):\[ 30^2 - 4 \cdot 25 \cdot 9 = 900 - 900 = 0 \]

Step 5: Solve for q

Since the discriminant is 0, there is only one solution for \(q\):\[ q = \frac{-30 \pm 0}{50} = \frac{-30}{50} = -\frac{3}{5} \]

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the general form: \[ ax^2 + bx + c = 0 \] In this equation:

• represents the coefficient of the squared term
• b represents the coefficient of the linear term
• c is the constant term

The quadratic equation is called 'quadratic' because it involves a variable squared (i.e., raised to the power of 2). Solving a quadratic equation means finding the values of the variable that make the equation true.

Coefficients

Coefficients are the numerical factors in the terms of an equation. For the quadratic equation \[ ax^2 + bx + c = 0 \],

• a, b, and c are the coefficients

For example, in the quadratic equation\[ 25q^2 + 30q + 9 = 0 \]

• a = 25
• b = 30
• c = 9

These coefficients play a crucial role when solving the equation using methods like the Quadratic Formula.

Discriminant

The discriminant of a quadratic equation gives important information about the nature of its solutions. It is part of the quadratic formula and is calculated as follows: \[ b^2 - 4ac \]In the quadratic equation\[ 25q^2 + 30q + 9 = 0 \],

• b = 30
• a = 25
• c = 9

So the discriminant is:\[ 30^2 - 4(25)(9) = 900 - 900 = 0 \]A discriminant of 0 indicates that the quadratic equation has exactly one distinct solution.

Single Solution

When solving a quadratic equation, the discriminant helps us determine how many solutions exist:

• If the discriminant is greater than 0, there are two real and distinct solutions.
• If the discriminant is equal to 0, there is exactly one real solution.
• If the discriminant is less than 0, there are no real solutions, only complex ones.

In the example \[ 25q^2 + 30q + 9 = 0 \]with a discriminant of 0, the equation \[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]reduces to \[ q = \frac{-30 \pm 0}{50} = -\frac{3}{5} \]
Hence, the quadratic equation has a single solution: \[ q = -\frac{3}{5} \]