Problem 32 - Algebra 1

Question

LOGICAL REASONING Consider the equation \(a x^{2}+b x+c=0\) and use the quadratic formula to justify the statement. If \(b^{2}-4 a c\) is positive, then the equation has two solutions.

Step-by-Step Solution

Answer

The statement is true. When \(b^{2}-4ac > 0\), the quadratic equation has two distinct real solutions given by the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\).

Step 1: Recollect the Quadratic Formula

The Quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). We're using \(b^{2}-4ac\) which is the discriminant.

Step 2: Check the value of the discriminant

The task is to prove that if \(b^{2}-4ac\) is positive, then the quadratic equation has two solutions. This involves understanding what happens when the discriminant is more than zero.

Step 3: Analyze the quadratic formula when discriminant is positive

When \(b^{2}-4ac > 0\), the expression under the square root in the quadratic formula will be a positive number. This means, you'll get two distinct numbers when you perform 'plus or minus' operation and thus get two different solutions to the quadratic equation.

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool used to find the solutions of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are coefficients of the equation.
The formula uses three parts:

• \(-b\): This part shifts all solutions.
• \(\pm \): This means there are potentially two solutions.
• \(\sqrt{b^2 - 4ac}\): The root determines the nature of the solutions.

The essence of the quadratic formula lies in evaluating the term under the square root, known as the discriminant. Understanding the quadratic formula makes it easier to grasp how the solutions to the quadratic equation are determined. It provides a systematic way to calculate roots accurately and is widely used due to its reliability.

Discriminant

The discriminant is a key part of the quadratic formula, represented as \( b^2 - 4ac \). It provides vital insight into the nature of the roots of the quadratic equation:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), the equation has two distinct real solutions.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), there is exactly one real solution, also known as a repeated or double root.
• If the discriminant is negative, \( b^2 - 4ac < 0 \), there are no real solutions; the solutions are complex or imaginary.

The discriminant helps predict the type and number of solutions before actually calculating them. This can be particularly useful in algebra to assess whether solving the equation is practical or to anticipate the nature of roots one might find. Knowing that the equation can have zero, one, or two solutions based on the sign of the discriminant streamlines many mathematical processes.

Solutions of Quadratic Equation

The solutions of a quadratic equation are the values of \( x \) that make the equation \( ax^2 + bx + c = 0 \) true. Depending on the discriminant, the nature and number of solutions change:

Two Solutions

When the discriminant is positive, the equation produces two distinct real solutions. This happens because the expression \( \pm \sqrt{b^2 - 4ac} \) gives two different results.

One Solution

If the discriminant is zero, there is a perfect square under the square root, leading to a single, repeated real solution. This can be thought of as the point where the parabola touches the x-axis.

Complex Solutions

When the discriminant is negative, the equation has no real solutions, and the solutions are complex numbers noted by the imaginary unit \( i \). This situation arises when the parabola does not intersect the x-axis at any point.
Understanding these solutions helps in visualizing the graph of a quadratic function and predicting its behavior. It illustrates how algebraic theories translate into graphical representations and helps in solving real-world problems where quadratic equations are applied.