Problem 91 - Precalculus Student Solutions Manual 5th

Question

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$8 x^{2}-72=0$$

Step-by-Step Solution

Answer

The equation has two distinct real rational solutions.

Step 1: Write the equation in standard form

The given equation is \[8x^2 - 72 = 0\]Rewrite it in the form \[ax^2 + bx + c = 0\]The standard form of the equation is: \[8x^2 + 0x - 72 = 0\]Here, \(a = 8\), \(b = 0\), and \(c = -72\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) is given by the formula: \[\Delta = b^2 - 4ac\]Substitute the values of \(a\), \(b\), and \(c\): \[\Delta = 0^2 - 4(8)(-72)\]\[\Delta = -4(8)(-72)\]\[\Delta = 4 \cdot 576\]\[\Delta = 2304\]

Step 3: Predict the number of distinct solutions

If \(\Delta > 0\), there are two distinct real solutions. Since \(\Delta = 2304 > 0\), the equation has two distinct real solutions.

Step 4: Determine the nature of the solutions

Since \(\Delta\) is a positive perfect square (\(2304 = 48^2\)), the two solutions are rational.

Key Concepts

quadratic equations

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \).
Quadratic equations are called 'quadratic' because they include terms up to the square of the variable (\(x^2\)).
Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These equations can be solved using various methods, including:

• Factoring
• Using the Quadratic Formula: \[ x = \frac{-b \, \text{±} \, \root \root 2202675,number_8692542988524719362,-4ac\root 2202675,number_8692541156192184320,2}} tu \root 2202676,number_8692542988524719362, 0000 \root 2202676,number_ 899467substrate< \]
• Completing the square
• Graphing

Quadratic equations can have different kinds of solutions based on the value of the discriminant.

real solutions

Real solutions to quadratic equations are the values for \(x\) that satisfy the quadratic equation and are real numbers.
These values are not complex or imaginary. The discriminant (\bold{Δ}) helps predict the number and types of real solutions:

• If \(\bold{Δ} > 0\), the equation has two distinct real solutions
• If \(\bold{Δ} = 0\), the equation has exactly one real solution (also called a double root or repeated root)
• If \(\bold{Δ} < 0\), the equation has no real solutions, but two non-real complex solutions

The equation \(8x^2 - 72 = 0 \) has a discriminant of 2304, which is greater than zero, indicating two distinct real solutions.

rational solutions

Rational solutions are specific types of real solutions. These solutions can be expressed as a ratio of two integers (fractions).
To determine if the solutions are rational, we again look at the discriminant:

• If the discriminant (\bold{Δ}) is a positive perfect square, the solutions are rational.
  Example: \(Δ= 2304\) and \(2304 = {48}^2\)
• If (\bold{Δ}) is positive but not a perfect square, the solutions are irrational (they cannot be represented as simple fractions)

Since \(Δ = 2304\) in our quadratic equation, and 2304 is a perfect square (\bold{48^2}), the solutions are rational.

nature of roots

The nature of roots of a quadratic equation depends on the discriminant (\bold{Δ = b^2 - 4ac}).
The discriminant tells us both the number of roots and their type:

• If \( Δ > 0 \), there are two distinct roots
  • If \( Δ = perfect \ square \), the roots are rational and distinct
  • If \( Δ eq perfect \ square \), the roots are irrational and distinct
• If \( Δ = 0\), there is a single root, often referred to as a double root, that is rational
• If \(\bold{Δ} < 0\), there are no real roots. Instead, there are two complex roots

For our specific equation \(8x^2 - 72 = 0\), the discriminant is 2304, a perfect square, meaning there are two distinct, rational roots.