Problem 18 - Algebra 1: Concepts and Skills

Question

Find the discriminant of the quadratic equation. \(7-5 x^{2}+9 x=x\)

Step-by-Step Solution

Answer

The discriminant of the quadratic equation is 204.

Step 1: Convert to Standard Form

Firstly, rearrange the given equation into its standard form. Subtract \(x\) from both sides to get: \(-5x^{2} + 9x + (7 - x) = 0\) which simplifies to: \(-5x^{2} + 8x + 7 = 0\) In this equation, \(a = -5\), \(b = 8\), and \(c = 7\)

Step 2: Calculate the Discriminant

Use formula \(D = b^{2} - 4ac\) to determine the discriminant: Substituting \(a = -5\), \(b = 8\), and \(c = 7\) into the formula: \(D = (8)^{2} - 4*(-5)*7 = 64 - (-140) = 64 + 140 = 204\)

Step 3: Interpret the Discriminant

Since the discriminant is greater than zero (\(D > 0\)), the quadratic equation has two distinct real roots.

Key Concepts

Standard Form of a Quadratic Equation

In algebra, the standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \) cannot be zero. This form is widely used because it clearly displays the highest degree of the variable, which is 2, indicating that the equation is quadratic. Furthermore, this format is conducive to various methods of solving quadratic equations, including factoring, completing the square, using the quadratic formula, and graphing.

The equation \( 7 - 5x^2 + 9x = x \) can be transformed into standard form by arranging the terms in decreasing power of \( x \) and ensuring the \( x^2 \) term is positive. This results in the equation \( -5x^2 + 8x + 7 = 0 \) with \( a = -5 \), \( b = 8 \), and \( c = 7 \). Simplifying and rewriting quadratic equations into standard form is a crucial step before proceeding to further analysis like calculating the discriminant or finding the roots.

Calculating the Discriminant

The discriminant of a quadratic equation is a key feature that can be calculated using the formula \( D = b^2 - 4ac \), where \( b \) and \( c \) are the coefficients from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \).

The value of the discriminant provides critical information about the nature of the roots without actually solving for them. In the case of our example, the discriminant is calculated by substituting the values of \( a \) as -5, \( b \) as 8, and \( c \) as 7 into the formula. This results in \( D = 8^2 - 4(-5)(7) = 64 + 140 = 204 \), a positive number. Understanding how to calculate the discriminant is essential as it serves as a quick-check tool for predicting the number and type of solutions to the quadratic equation.

Interpreting the Discriminant

Once the discriminant is calculated, its value can give us insight into the nature and number of solutions for the quadratic equation. If the discriminant is positive, as it is in our example with \( D = 204 \), it means there are two distinct real roots to the equation. This implies that if \( D > 0 \), the parabola crosses the x-axis at two points, and thus the quadratic equation will have two real and unequal solutions.

Alternatively, if \( D = 0 \), the equation has one real root, also known as a repeated root because the parabola only touches the x-axis at one point. And if \( D < 0 \), there are no real solutions since the parabola does not intersect the x-axis at all, suggesting that the solutions are complex numbers. The ability to interpret the discriminant is a valuable skill in algebra as it allows students to anticipate the number and types of solutions that a quadratic equation may have without solving the equation completely.