Problem 73 - Algebra and Trigonometry

Question

Solve the equation for \(x\) $$ a^{2} x^{2}+2 a x+1=0 \quad(a \neq 0) $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{a}\).

Step 1: Understand the Quadratic Equation

The equation given is a quadratic equation in the form \(ax^2 + bx + c = 0\). For this problem, \(a = a^2\), \(b = 2a\), and \(c = 1\). We need to solve for \(x\).

Step 2: Identify the Coefficients

Identify the coefficients for the quadratic formula. Here, we have: - \(a\) (coefficient of \(x^2\)) as \(a^2\)- \(b\) (coefficient of \(x\)) as \(2a\)- \(c\) (constant term) as \(1\).

Step 3: Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Substituting the coefficients:\(x = \frac{-2a \pm \sqrt{(2a)^2 - 4(a^2)(1)}}{2a^2}\).

Step 4: Simplify Inside the Square Root

Calculate the discriminant \(b^2 - 4ac\):\[(2a)^2 - 4 imes a^2 imes 1 = 4a^2 - 4a^2 = 0\]So the discriminant is \(0\).

Step 5: Solve for x with Zero Discriminant

When the discriminant is 0, there is exactly one real solution to the equation:\[x = \frac{-2a + 0}{2a^2} = \frac{-2a}{2a^2} = \frac{-1}{a}\].

Step 6: Conclude the Solution

The solution to the quadratic equation is \(x = -\frac{1}{a}\). Since the discriminant is 0, this is the only solution.

Key Concepts

Quadratic Formula

To solve any quadratic equation of the form \( ax^2 + bx + c = 0 \), we can use the quadratic formula. This handy formula is written as:

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

It helps us find the values of \( x \) by providing solutions derived from the equation's coefficients. The formula includes a square root and division, which will be determined by the specific parts within the equation.
By substituting the coefficients into the formula, we can efficiently solve for the variable \( x \). It's crucial to carefully identify these coefficients to ensure accuracy in computing the solution.

Discriminant

In the quadratic formula, the expression under the square root \( b^2 - 4ac \) is known as the discriminant. This is a vital part of solving quadratic equations because:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If negative, there are no real solutions, only complex ones.

In our exercise, the discriminant \( b^2 - 4ac \) simplifies to 0, indicating that there is exactly one real solution for the equation. The discriminant tells us about the nature and number of solutions without solving fully, making it an excellent tool for checking our work.

Coefficients

Coefficients are the numbers in front of the variables in a polynomial equation like the quadratic. For example:

• In \( ax^2 + bx + c = 0 \), \( a \), \( b \), and \( c \) are the coefficients.

They tell you how much each term in the polynomial is "scaled". In the given exercise, the coefficients are \( a^2 \) for \( x^2 \), \( 2a \) for \( x \), and \( 1 \) as the constant term.
Understanding the role and value of these coefficients is key to using the quadratic formula correctly and solving the equation accurately.

Real Solutions

Real solutions are outcomes of quadratic equations that are real numbers. These arise directly from the quadratic formula:

• If the discriminant is \( b^2 - 4ac = 0 \), there's one real solution.
• If it's positive, we find two distinct real solutions, \( x_1 \) and \( x_2 \).

In our specific problem, the discriminant is zero, leading to only one real solution.
The equation confirms \( x = -\frac{1}{a} \) as the only real solution. Real solutions indicate points where the quadratic equation intersects the x-axis on a graph, showing where the roots or zeroes of the expression lie.