Problem 133 - Elementary Algebra

Question

Determine the number of solutions to each quadratic equation. (a) \(r^{2}+12 r+36=0\) \(8 t^{2}-11 t+5=0\) \(4 u^{2}-12 u+9=0\) \(3 v^{2}-5 v-1=0\)

Step-by-Step Solution

Answer

(a) one solution, 8t^2 - 11t + 5 = 0: no real solutions, 4u^2 - 12u + 9 = 0: one solution, 3v^2 - 5v - 1 = 0: two solutions.

Step 1: Determine the Discriminant for (a) \(r^2 + 12r + 36 = 0\)

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). For \(r^2 + 12r + 36 = 0\), \(a = 1\), \(b = 12\), and \(c = 36\). Calculate the discriminant: \(D = 12^2 - 4(1)(36) = 144 - 144 = 0\).

Step 2: Determine the Number of Solutions for (a)

When the discriminant \(D = 0\), the quadratic equation has exactly one real solution.

Step 3: Determine the Discriminant for \(8t^2 - 11t + 5 = 0\)

For \(8t^2 - 11t + 5 = 0\), identify the coefficients: \(a = 8\), \(b = -11\), and \(c = 5\). Calculate the discriminant: \(D = (-11)^2 - 4(8)(5) = 121 - 160 = -39\).

Step 4: Determine the Number of Solutions for \(8t^2 - 11t + 5 = 0\)

When the discriminant \(D < 0\), the quadratic equation has no real solutions, but two complex solutions.

Step 5: Determine the Discriminant for \(4u^2 - 12u + 9 = 0\)

For \(4u^2 - 12u + 9 = 0\), identify the coefficients: \(a = 4\), \(b = -12\), and \(c = 9\). Calculate the discriminant: \(D = (-12)^2 - 4(4)(9) = 144 - 144 = 0\).

Step 6: Determine the Number of Solutions for \(4u^2 - 12u + 9 = 0\)

When the discriminant \(D = 0\), the quadratic equation has exactly one real solution.

Step 7: Determine the Discriminant for \(3v^2 - 5v - 1 = 0\)

For \(3v^2 - 5v - 1 = 0\), identify the coefficients: \(a = 3\), \(b = -5\), and \(c = -1\). Calculate the discriminant: \(D = (-5)^2 - 4(3)(-1) = 25 + 12 = 37\).

Step 8: Determine the Number of Solutions for \(3v^2 - 5v - 1 = 0\)

When the discriminant \(D > 0\), the quadratic equation has two distinct real solutions.

Key Concepts

Discriminant

The discriminant is a key concept when working with quadratic equations. It's represented by the symbol \( D \) and is calculated using the formula: \[ D = b^2 - 4ac \] The coefficients \(a\), \(b\), and \(c\) come from the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). The discriminant helps us determine the nature of the solutions (roots) of the equation. Calculating the discriminant gives us an idea about whether the solutions are real or complex.

For example, for the equation \( r^2 + 12r + 36 = 0 \), we identify the coefficients as \(a = 1\), \(b = 12\), and \(c = 36\). Plugging these values into the discriminant formula, we get \( D = 12^2 - 4(1)(36) = 144 - 144 = 0 \).

The discriminant can have three types of values: positive, negative, or zero. Each type indicates different kinds of solutions.

Number of Solutions

The discriminant directly informs us about the number of solutions to a quadratic equation. Understanding this concept is essential for solving and predicting the behavior of the equation.

The number of solutions is determined as follows:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution.
• If \( D < 0 \), there are two complex solutions with no real parts.

For example, in the equation \( 8t^2 - 11t + 5 = 0 \), the coefficients are \(a = 8\), \(b = -11\), and \(c = 5\). The discriminant is \( D = (-11)^2 - 4(8)(5) = 121 - 160 = -39 \). Since \( D < 0 \), this equation has two complex solutions.

Real and Complex Solutions

Solutions to quadratic equations can be real or complex, and which type you get depends on the discriminant.

• Real Solutions: When the discriminant \( D \) is greater than or equal to zero, the solutions are real.
  
  For example, for the equation \( 4u^2 - 12u + 9 = 0 \), the discriminant is \( D = (-12)^2 - 4(4)(9) = 144 - 144 = 0 \). Since \(D = 0\), this equation has exactly one real solution.
• Complex Solutions: When the discriminant \( D \) is less than zero, the solutions are complex and involve imaginary numbers.
  
  For the equation \( 3v^2 - 5v - 1 = 0 \), the discriminant is \( D = (-5)^2 - 4(3)(-1) = 25 + 12 = 37 \). Since \( D > 0 \), this equation has two distinct real solutions.

By knowing the discriminant, you can quickly determine what type of solutions a quadratic equation will have, making it easier to solve and understand the nature of the equation.