Problem 44 - Algebra for College Students

Question

Find \(b^{2}-4 a c\) and the number of real solutions to each equation. $$12-7 x+x^{2}=0$$

Step-by-Step Solution

Answer

The discriminant is 1 and there are two distinct real solutions.

Step 1: Identify coefficients

Rewrite the quadratic equation in standard form: a x^{2} + b x + c = 0. Here, rearrange and rewrite the equation: x^{2} - 7x + 12 = 0. Identify the coefficients: a = 1, b = -7, c = 12.

Step 2: Calculate the discriminant

Use the formula for the discriminant b^{2} - 4ac. Substitute the values of a, b, and c into the formula: b^{2} - 4ac = (-7)^{2} - 4(1)(12) = 49 - 48.

Step 3: Simplify the discriminant

Simplify the expression obtained in the previous step: b^{2} - 4ac = 49 - 48 = 1.

Step 4: Determine the number of real solutions

Use the value of the discriminant to determine the number of real solutions: If b^{2} - 4ac > 0, the equation has two distinct real solutions. If b^{2} - 4ac = 0, the equation has exactly one real solution. If b^{2} - 4ac < 0, the equation has no real solutions. Since b^{2} - 4ac = 1 > 0, the equation x^{2} - 7x + 12 = 0 has two distinct real solutions.

Key Concepts

discriminant

In a quadratic equation, the discriminant is a key concept to understand. The discriminant of a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\). This value determines the nature of the roots of the quadratic equation. The discriminant gives insight into whether the solutions are real or complex numbers and how many solutions there are. To find the discriminant in the given problem, we first need to rewrite the equation \(12 - 7x + x^2 = 0\) in its standard form, so it becomes \(x^2 - 7x + 12 = 0\). From the standard form, we identify the coefficients as:

• a = 1
• b = -7
• c = 12

Substituting these values into the discriminant formula gives: \[(-7)^2 - 4(1)(12) = 49 - 48 = 1\]
As we can see, the discriminant (\(1\)) is positive.

real solutions

The discriminant tells us about the number of real solutions for the quadratic equation. Here's how to interpret the value:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution (a repeated or double root).
• If \(b^2 - 4ac < 0\), there are no real solutions (the solutions are complex numbers).

For the equation \(x^2 - 7x + 12 = 0\), we calculated the discriminant to be 1. Since \(1 > 0\), this means the quadratic equation has two distinct real solutions. This tells us that the graph of this equation will intersect the x-axis at two different points.

coefficients

Coefficients in a quadratic equation are the numerical factors of the terms. For the quadratic equation in standard form \(ax^2 + bx + c = 0\), the coefficients are

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

In our problem, the equation \(12 - 7x + x^2 = 0\) is rewritten in standard form as \(x^2 - 7x + 12 = 0\). Accordingly, the coefficients are:

• a = 1 (coefficient of \(x^2\))
• b = -7 (coefficient of \(x\))
• c = 12 (constant term)

Identifying these coefficients is crucial because they are used in the discriminant formula \(b^2 - 4ac\) to determine the nature and number of solutions to the quadratic equation.