Problem 43 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 4 x^{2}+16 x-9=0 $$

Step-by-Step Solution

Answer

Solutions are \( x = \frac{1}{2} \) and \( x = -\frac{9}{2} \).

Step 1: Identify the Quadratic Equation Format

The given equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = 4 \), \( b = 16 \), and \( c = -9 \).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). Substitute the values: \( 16^2 - 4 \times 4 \times (-9) \). This simplifies to \( 256 + 144 = 400 \).

Step 3: Determine the Nature of the Roots

The discriminant is positive (\( 400 > 0 \)), which means the equation has two distinct real solutions.

Step 4: Apply the Quadratic Formula

The solutions to the quadratic equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 4 \), \( b = 16 \), \( c = -9 \), and \( \sqrt{400} = 20 \).

Step 5: Calculate Each Root

Calculate the two solutions: 1. \( x_1 = \frac{-16 + 20}{8} = \frac{4}{8} = \frac{1}{2} \).2. \( x_2 = \frac{-16 - 20}{8} = \frac{-36}{8} = -\frac{9}{2} \).

Key Concepts

Real Solutions

In the context of quadratic equations, real solutions refer to the values of the variable \( x \) that satisfy the equation and are real numbers. When you solve a quadratic equation of the form \( ax^2 + bx + c = 0 \), the solutions can be real or complex. For an equation to have real solutions, it is determined by calculating the discriminant, a crucial part of the quadratic formula.
The real solutions for the given equation \( 4x^2 + 16x - 9 = 0 \) can be found by solving using the quadratic formula. Since the discriminant of this equation is positive, both solutions are real numbers, meaning they can be plotted on the number line. In this case, the equation has two distinct real solutions, which we calculated as \( \frac{1}{2} \) and \(-\frac{9}{2} \).
Real solutions mean that the graph of the quadratic function intersects the \( x \)-axis at these points. This aspect is crucial for understanding how the values of \( x \) relate to the nature of the solutions.

Discriminant

The discriminant is the part of the quadratic equation formula that determines the nature of the roots of any quadratic equation. It is represented by \( b^2 - 4ac \), which is derived from the standard quadratic equation \( ax^2 + bx + c = 0 \). The value of the discriminant helps us to predict how many solutions there are and what type they will be.
For the equation \( 4x^2 + 16x - 9 = 0 \), we calculated the discriminant as follows:

• \( b^2 = 16^2 = 256 \)
• \( 4ac = 4 \times 4 \times (-9) = -144 \)
• Thus, the discriminant is \( 256 + 144 = 400 \)

A positive discriminant (as in this case) signals that there are two distinct real solutions. A discriminant of zero would indicate exactly one real solution. A negative discriminant would imply no real solutions, but rather two complex solutions.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of any quadratic equation. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula solves for \( x \) by taking into account the coefficients \( a \), \( b \), and \( c \) in the quadratic equation \( ax^2 + bx + c = 0 \).
Using our equation \( 4x^2 + 16x - 9 = 0 \), we apply the quadratic formula:

• \( a = 4 \), \( b = 16 \), and \( c = -9 \)
• \( \sqrt{b^2 - 4ac} = \sqrt{400} = 20 \)
• Therefore, the solutions are:
  • \( x_1 = \frac{-16 + 20}{8} \)
  • \( x_2 = \frac{-16 - 20}{8} \)

The quadratic formula is a universal solution method because it delivers the roots without needing to factorize the equation, which can sometimes be challenging.

Roots

Roots of a quadratic equation refer to the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These are essentially the solutions we calculate. Each root represents a point where the graph of the equation intersects the \( x \)-axis.
For the equation \( 4x^2 + 16x - 9 = 0 \), the roots are calculated using the quadratic formula. We found:

• \( x_1 = \frac{1}{2} \)
• \( x_2 = -\frac{9}{2} \)

Both values are real, confirming they are points where the parabola intersects the \( x \)-axis. Remember, identifying roots helps us understand the behavior of a quadratic function, its direction, and its intercepts with the axes.

Nature of Roots

The nature of roots in a quadratic equation gives us insight into whether the solutions are real and how many there are. This is primarily determined by the discriminant \( b^2 - 4ac \), which indicates whether the solutions are real or complex numbers.
For the equation \( 4x^2 + 16x - 9 = 0 \), a discriminant of 400 indicates:

• Since 400 is positive, the equation has two distinct real roots.
• If it were zero, there would be a single real root (a repeated root).
• A negative value would mean two complex roots.

Thus, understanding the nature of roots is vital when predicting how the quadratic function behaves and what type of solutions you can expect.