Problem 67 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the discriminant to determine the number of real solutions of the equation. $$ 4 x^{2}+12 x+9=0 $$

Step-by-Step Solution

Answer

The given quadratic equation is \(4x^2 + 12x + 9 = 0\), with coefficients \(a = 4\), \(b = 12\), and \(c = 9\). Using the discriminant formula \(\Delta = b^2 - 4ac\), we get \(\Delta = 144 - 144 = 0\). Since the discriminant is 0, there is one real solution for the given equation.

Step 1: Identify coefficients a, b, and c

From the given quadratic equation \(4x^2 + 12x + 9 = 0\), we can identify the coefficients as follows: \(a = 4\), \(b = 12\), and \(c = 9\).

Step 2: Calculate the discriminant

Using the discriminant formula \(\Delta = b^2 - 4ac\), let's plug in the values of a, b, and c, and calculate the discriminant: \[ \Delta = 12^2 - 4 \times 4 \times 9 \]

Step 3: Simplify the discriminant

Continue to simplify the discriminant calculation: \[ \Delta = 144 - 144 \\ \Delta = 0 \]

Step 4: Interpret the discriminant

The discriminant is equal to 0, which implies that there is one real solution for the given quadratic equation. So, the given equation \(4x^2 + 12x + 9=0\) has one real solution.

Key Concepts

Real Solutions of Quadratic Equations

Understanding the nature of the solutions for quadratic equations is essential for students as it extends to various applications in mathematics and sciences. A quadratic equation follows the standard form of
\( ax^2 + bx + c = 0 \), with real coefficients a, b, and c, and where \(a eq 0\).
The solutions to a quadratic equation—where 'solution' means the values of x that satisfy the equation—are found where the quadratic graph intersects the x-axis. These intersection points are known as the roots, and they can be real or complex. When we say 'real solutions', we're referring to roots that are real numbers.
If the discriminant, denoted as \(\triangle\), is positive, the quadratic equation has two distinct real roots. If the discriminant is zero, there is exactly one real root, also known as a repeated or double root. However, if the discriminant is negative, the equation has no real roots but instead has two complex roots.

Calculating Discriminant

The discriminant is a powerful tool in determining the number and nature of solutions for a quadratic equation without actually solving the equation. It is calculated using the formula

\( \triangle = b^2 - 4ac \),
where 'a' is the coefficient in front of \( x^2 \), 'b' the coefficient in front of x, and 'c' is the constant term.

• If \( \triangle > 0 \), the quadratic equation has two distinct real solutions.
• If \( \triangle = 0 \), there is exactly one real solution.
• If \( \triangle < 0 \), the solutions are complex and there are no real solutions.

The value of the discriminant reveals not just the quantity but also the quality of the roots, describing whether they are rational and irrational.

Quadratic Formula

The quadratic formula is a fundamental formula for solving quadratic equations and directly finding the roots, given by

\( x = \frac{{-b \[\pm\] \sqrt{{\triangle}}}}{{2a}} \),
where \( \triangle \) is the discriminant previously discussed. This formula can be used irrespective of the discriminant's value but is particularly useful in finding exact forms of the roots when the discriminant is not a perfect square. The \( \[\pm\] \) symbol indicates that the quadratic equation might have two solutions which are obtained by adding and subtracting the square root of the discriminant respectively.

Coefficient Identification in Quadratics

Identifying the coefficients in a quadratic equation is the starting point for analyzing its solutions. Coefficients are simply the numerical or literal multipliers of the terms in the equation. In the standard form \( ax^2 + bx + c = 0 \), 'a' is the coefficient of \( x^2 \), 'b' is the coefficient of x, and 'c' is the constant term not associated with x.
These coefficients directly influence the graph's shape and position, and they are crucial for determining the discriminant and using the quadratic formula. Accurate identification ensures the proper application of the formula and correct calculation of the discriminant, thereby leading to a successful solution of the quadratic equation.