Problem 33 - 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 zero, then the equation has one solution.

Step-by-Step Solution

Answer

The statement 'If \(b^{2}-4 a c\) is zero, then the equation has one solution.' is true as per the definition of discriminant in the Quadratic formula.

Step 1: Understanding the Quadratic Formula

The quadratic formula is used to solve quadratic equations and is as follows: \(x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\) Where \(a\), \(b\), and \(c\) are coefficients of the equation \(a x^{2} + b x + c = 0\), and \(x\) are the solutions to the equation. The expression under the square root, \(b^{2}-4 a c\), is called discriminant.

Step 2: Conditions on the discriminant

The discriminant in the quadratic formula, \(b^{2}-4 a c\), determines the number and nature of solutions to the quadratic equation. If the discriminant is greater than zero, the equation has two real solutions; if it is equal to zero, the equation has one real solution; and if it is less than zero, the equation has two complex solution.

Step 3: Evaluating The Statement

The provided statement says 'If \(b^{2}-4 a c\) is zero, then the equation has one solution.' From Step 2, we can verify that this statement is true. If the discriminant is zero, then the square root term in the quadratic formula disappears resulting in one real solution.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula allows you to find the solutions, also known as the roots, of the equation. The quadratic formula is expressed as:

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

Here, \(a\), \(b\), and \(c\) are coefficients of the quadratic equation. The value of \(x\) calculated using this formula gives you the points where the equation equals zero. This formula is essential in algebra and helps determine how many solutions an equation has, whether they're real or complex.

Discriminant

At the heart of the quadratic formula, you'll find the discriminant, denoted as \(b^2 - 4ac\). This part of the formula holds crucial insights into the nature and number of solutions.The discriminant serves as a kind of compass:

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

Understanding the discriminant is vital for predicting how many times a quadratic curve will intersect the x-axis, if at all.

Real Solutions

Real solutions occur when a quadratic equation's discriminant (\(b^2 - 4ac\)) is zero or positive.

• If \(b^2 - 4ac > 0\), you have two different real solutions. These solutions indicate two points where the parabola crosses the x-axis.
• If \(b^2 - 4ac = 0\), there is exactly one real solution. This case represents the parabola touching the x-axis at a single point, known as the vertex.

Real solutions are significant because they indicate where the output of the quadratic function reaches zero, a key concept in graphing or analyzing quadratic equations.

Complex Solutions

When a quadratic equation has no real solutions, it means the discriminant \(b^2 - 4ac\) is less than zero. That means the parabola does not intersect the x-axis at any point. Instead, the solutions of the equation are complex numbers, which involve the imaginary unit \(i\).The solutions can be written as:

• \[x = \frac{-b \pm i\sqrt{|b^2 - 4ac|}}{2a}\]

Where \(i\) represents the square root of negative one. Complex solutions are essential in fields like engineering and physics, where processes might involve oscillations or wave patterns. Understanding complex solutions allows exploration of mathematical problems beyond the real number line.