Problem 46 - Algebra A Combined Function

Question

Use the discriminant to determine the number and types of solutions of each equation. $$ 3 x^{2}=5-7 x $$

Step-by-Step Solution

Answer

The equation has 2 distinct real solutions.

Step 1: Write the equation in standard form

To use the discriminant, the equation must be in standard quadratic form, which is a format like \( ax^2 + bx + c = 0 \). Start by moving all terms to one side of the equation.The original equation is:\[ 3x^2 = 5 - 7x \]Rearrange it as follows:\[ 3x^2 + 7x - 5 = 0 \]

Step 2: Identify coefficients

In the standard form \( ax^2 + bx + c = 0 \), identify the coefficients:- \( a = 3 \)- \( b = 7 \)- \( c = -5 \)

Step 3: Calculate the discriminant

The discriminant \( \, \Delta\, \) is calculated using the formula:\[ \Delta = b^2 - 4ac \]Substitute the identified coefficients into this formula:\[ \Delta = 7^2 - 4 \times 3 \times (-5) \]\[ \Delta = 49 + 60 \]\[ \Delta = 109 \]

Step 4: Determine the number and type of solutions

The discriminant helps determine the number and types of solutions of the quadratic equation:- If \( \Delta > 0 \), there are 2 distinct real solutions.- If \( \Delta = 0 \), there is 1 real solution (a repeated root).- If \( \Delta < 0 \), there are no real solutions (two complex solutions).Since \( \Delta = 109 > 0 \), the equation has 2 distinct real solutions.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are polynomial equations of degree 2, typically expressed in the form \( ax^2 + bx + c = 0 \). This means they have a variable raised to the second power as their highest degree term. Quadratic equations can have up to two solutions since the graph of a quadratic equation is a parabola, which can intersect the x-axis at zero, one, or two points depending on the equation:

• If the parabola intersects the x-axis at two distinct points, it means two real solutions exist.
• If it touches the x-axis at a single point, that implies a repeated real solution.
• If it doesn't intersect at all, the solutions are complex numbers.

These properties make quadratic equations widely applicable in fields such as physics, engineering, and economics. Recognizing the pattern and structure of these equations is crucial when solving them.

Identifying Real Solutions through the Discriminant

The discriminant of a quadratic equation is a special formula used to determine the nature and number of solutions. It is calculated using the formula \( \Delta = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients of the equation \( ax^2 + bx + c = 0 \). The value of the discriminant provides critical information about the solutions:

• If \( \Delta > 0 \), the quadratic equation has two distinct real solutions.
• If \( \Delta = 0 \), there is one real solution, which means the roots are repeated.
• If \( \Delta < 0 \), the equation has no real solutions but rather two complex solutions.

Understanding how to compute and interpret the discriminant is essential in algebra as it provides a quick way to assess whether solutions to the equation can exist in the real number system.

Standard Form of Quadratic Equations

The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). This form is crucial because it sets the stage for many common methods of solution, including factoring, using the quadratic formula, and completing the square. Transforming any quadratic equation into this form ensures that the equation is ready for analysis or additional solving techniques.Consider the equation \( 3x^2 = 5 - 7x \). To bring it into standard form, all terms must be moved to one side of the equation and combined:- Moving all terms gives us \( 3x^2 + 7x - 5 = 0 \).With the quadratic equation in standard form, we can now effectively identify coefficients and subsequently apply the discriminant or other methods to solve the equation.

Role of Coefficients in Quadratic Equations

Coefficients are the numbers positioned in front of variables in a polynomial equation. In the context of a quadratic equation \( ax^2 + bx + c = 0 \):

• \( a \) is the leading coefficient and determines the opening direction and width of the parabola (upwards if positive, downwards if negative).
• \( b \) affects the position of the vertex and symmetry of the parabola.
• \( c \) is the constant term and represents the y-intercept of the parabola where the graph crosses the y-axis.

After transforming the example equation \( 3x^2 = 5 - 7x \) into \( 3x^2 + 7x - 5 = 0 \), the coefficients are identified as:- \( a = 3 \)- \( b = 7 \)- \( c = -5 \)Knowing these coefficients is essential. They pave the way not only for computing the discriminant but also for applying other algebraic methods to solve the quadratic equation.