Problem 38 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 8 x^{2}-6 x-9=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3}{2} \) and \( x = \frac{-3}{4} \).

Step 1: Identify the Quadratic Equation

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \). Here, \( a = 8 \), \( b = -6 \), and \( c = -9 \).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated as \( b^2 - 4ac \). Substitute the values to find \( \Delta = (-6)^2 - 4 \times 8 \times (-9) = 36 + 288 = 324 \).

Step 3: Evaluate the Discriminant

Since the discriminant \( \Delta = 324 \) is positive, the equation has two distinct real solutions.

Step 4: Use the Quadratic Formula

The solutions are given by the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \). Substitute \( b = -6 \), \( \Delta = 324 \), and \( a = 8 \) into the formula: \( x = \frac{6 \pm \sqrt{324}}{16} \).

Step 5: Calculate the Square Root

Compute \( \sqrt{324} = 18 \). So the solutions simplify to \( x = \frac{6 \pm 18}{16} \).

Step 6: Find the Two Solutions

Substitute the values into the formula:- For the plus sign: \( x = \frac{6 + 18}{16} = \frac{24}{16} = \frac{3}{2} \).- For the minus sign: \( x = \frac{6 - 18}{16} = \frac{-12}{16} = \frac{-3}{4} \).

Key Concepts

Discriminant of a Quadratic Equation

When solving a quadratic equation, the discriminant plays a crucial role. The discriminant is the part of the quadratic equation under the square root in the quadratic formula and is calculated using the formula: \( \Delta = b^2 - 4ac \). This small piece of the puzzle tells us how many real solutions we can expect from our quadratic equation.

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, also known as a repeated or double root.
• If \( \Delta < 0 \), no real solutions exist; instead, you have two complex solutions.

In the example equation \( 8x^2 - 6x - 9 = 0 \), the discriminant is calculated to be 324, which is positive. This indicates two distinct real solutions are possible.

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions of quadratic equations. The formula is given by:\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] It takes into account the coefficients from the standard quadratic form \( ax^2 + bx + c = 0 \):

• \( a \), the coefficient of \( x^2 \)
• \( b \), the coefficient of \( x \)
• \( c \), the constant term

Using the quadratic formula, you can quickly determine the roots of any quadratic equation, provided you know \( a \), \( b \), and \( c \). In our example, substituting the values \( a = 8 \), \( b = -6 \), and \( \Delta = 324 \), we solve for \( x \) and find:\[ x = \frac{-(-6) \pm 18}{16} \] This simplifies to \( x = \frac{6 \pm 18}{16} \), which gives the two potential solutions.

Real Solutions of Quadratic Equations

Real solutions of a quadratic equation refer to the x-values where the quadratic graph intersects the x-axis. They are called real because they represent actual points on a number line, as opposed to complex solutions, which involve imaginary numbers.
Once you've plugged values into the quadratic formula and calculated \( \Delta \), it's about performing some basic calculations:

• Calculate \( x = \frac{6 + 18}{16} = \frac{24}{16} = \frac{3}{2} \)
• Calculate \( x = \frac{6 - 18}{16} = \frac{-12}{16} = \frac{-3}{4} \)

Both results, \( \frac{3}{2} \) and \( \frac{-3}{4} \), are the real solutions for the equation \( 8x^2 - 6x - 9 = 0 \). They indicate the points where the parabola would cross the x-axis, confirming that the equation has two distinct real solutions due to the positive discriminant.