Problem 36 - Algebra for College Students

Question

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$-x^{2}+3 x-4=0$$

Step-by-Step Solution

Answer

Discriminant: -7. No real solutions.

Step 1 - Identify coefficients

The given quadratic equation is -x^{2}+3 x-4=0 Identify the coefficients as follows: a = -1, b = 3, c = -4

Step 2 - Calculate the discriminant

Use the formula for the discriminant: b^{2}-4ac Substitute the values of the coefficients: (3)^{2}-4(-1)(-4) = 9 - 16 Calculate the value: = -7

Step 3 - Determine the number of real solutions

Using the value of the discriminant, -7, determine the number of real solutions. If the discriminant is greater than 0, there are 2 real solutions; if it is equal to 0, there is 1 real solution; if it is less than 0, there are no real solutions. Since -7 is less than 0, there are no real solutions.

Key Concepts

discriminant

The discriminant is an essential part of solving quadratic equations. It helps us determine the nature of the roots without actually solving the equation. The formula for the discriminant (\textbf{D}) is given by \ \(b^2 - 4ac\).
In the equation \(-x^2 + 3x - 4 = 0\), we identify the coefficients as follows:
\(a = -1\), \(b = 3\), and \(c = -4\).
By substituting these values into the formula, we get:
\(3^2 - 4(-1)(-4)\) which simplifies to \(9 - 16\) giving us \(-7\).

So, in this case, the discriminant is \(-7\).

real solutions

Understanding the number of real solutions for a quadratic equation involves examining the value of the discriminant. The rule of thumb is:

• If the discriminant is greater than 0, the equation has two distinct real solutions.
• If the discriminant is equal to 0, it has exactly one real solution.
• If the discriminant is less than 0, the equation has no real solutions.

Based on our previous calculation, the discriminant of \(-7\) is less than 0.
Therefore, the quadratic equation \(-x^2 + 3x - 4 = 0\) has no real solutions.

coefficients

Coefficients are the numerical values in front of the variables in a polynomial expression. They play a crucial role in determining the shape and position of the graph of the equation.
In our equation \(-x^2 + 3x - 4 = 0\), the coefficients are:

• \(a = -1\) (coefficient of \(x^2\))
• \(b = 3\) (coefficient of \(x\))
• \(c = -4\) (constant term)

Identifying these values is essential for various calculations, such as finding the discriminant and understanding the nature of the roots. The signs and magnitudes of the coefficients can also give insights into the parabola's direction (opening upwards or downwards) and steepness.