Please correct the highlighted errors and try again.

'; return; } const a = parseFloat(document.getElementById('a').value); const b = parseFloat(document.getElementById('b').value); const c = parseFloat(document.getElementById('c').value); if (a === 0) { document.getElementById('result').innerHTML = '

The value of "a" cannot be zero for a quadratic equation.

'; return; } const discriminant = b * b - 4 * a * c; let steps = `

Step 1: Calculate the discriminant (b² - 4ac)

Discriminant = (${b}²) - 4 * ${a} * ${c} = ${discriminant}

`; if (discriminant > 0) { const root1 = (-b + Math.sqrt(discriminant)) / (2 * a); const root2 = (-b - Math.sqrt(discriminant)) / (2 * a); steps += `

Step 2: Since the discriminant is positive, there are two real and different roots.

Root 1 = (-b + √Discriminant) / (2a) = (-${b} + √${discriminant}) / (2 * ${a}) = ${root1.toFixed(2)}

Root 2 = (-b - √Discriminant) / (2a) = (-${b} - √${discriminant}) / (2 * ${a}) = ${root2.toFixed(2)}

`; document.getElementById('result').innerHTML = ``; } else if (discriminant === 0) { const root = -b / (2 * a); steps += `

Step 2: Since the discriminant is zero, there is one real and repeated root.

Root = -b / (2a) = -${b} / (2 * ${a}) = ${root.toFixed(2)}

`; document.getElementById('result').innerHTML = ``; } else { const realPart = (-b / (2 * a)).toFixed(2); const imaginaryPart = (Math.sqrt(-discriminant) / (2 * a)).toFixed(2); steps += `

Step 2: Since the discriminant is negative, there are two complex roots.

Root 1 = (-b / (2a)) + (i * √-Discriminant / (2a)) = ${realPart} + ${imaginaryPart}i

Root 2 = (-b / (2a)) - (i * √-Discriminant / (2a)) = ${realPart} - ${imaginaryPart}i

`; document.getElementById('result').innerHTML = ``; } document.getElementById('steps').innerHTML = steps; document.getElementById('toggleStepsBtn').style.display = 'block'; document.getElementById('copyResultBtn').style.display = 'block'; }); document.getElementById('resetBtn').addEventListener('click', function() { document.getElementById('a').value = ''; document.getElementById('b').value = ''; document.getElementById('c').value = ''; document.getElementById('result').textContent = ''; document.getElementById('steps').textContent = ''; document.getElementById('toggleStepsBtn').style.display = 'none'; document.getElementById('copyResultBtn').style.display = 'none'; document.getElementById('a-feedback').textContent = ''; document.getElementById('b-feedback').textContent = ''; document.getElementById('c-feedback').textContent = ''; }); document.getElementById('toggleStepsBtn').addEventListener('click', function() { const steps = document.getElementById('steps'); if (steps.style.display === 'none') { steps.style.display = 'block'; document.getElementById('toggleStepsBtn').textContent = 'Hide Steps'; } else { steps.style.display = 'none'; document.getElementById('toggleStepsBtn').textContent = 'Show Steps'; } }); document.getElementById('copyResultBtn').addEventListener('click', function() { const resultText = document.getElementById('result').textContent; navigator.clipboard.writeText(resultText).then(() => { alert('Result copied to clipboard'); }).catch(err => { console.error('Error copying text: ', err); }); });

Quadratic Formula Calculator: A Comprehensive Guide

The quadratic formula calculator is a powerful tool that helps solve quadratic equations, which are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the unknown variable. This guide will delve into the intricacies of the quadratic formula calculator, exploring its functionalities, applications, and usage.

• CPM CALCULATOR

Unveiling the Quadratic Formula

The quadratic formula, a mathematical expression used to determine the solutions (roots) of quadratic equations, is represented as:

x = (-b ± √(b² - 4ac)) / (2a)

where:

• a, b, and c: These are coefficients (numerical constants) in the quadratic equation (ax^2 + bx + c = 0). Note that ‘a’ cannot be zero, or the equation wouldn’t be considered quadratic.
• ± : This symbol represents “plus or minus.” There will be two possible solutions for x in most cases.
• √ : This symbol represents the square root.

Demystifying the Discriminant

The discriminant, an integral part of the quadratic formula, plays a crucial role in determining the nature of the solutions:

• Discriminant > 0: This indicates two distinct real number solutions for x.
• Discriminant = 0: This results in one repeated real number solution for x.
• Discriminant < 0: This signifies two complex number solutions for x (involving imaginary unit ‘i’).

Harnessing the Power of the Quadratic Formula Calculator

A quadratic formula calculator simplifies the process of solving quadratic equations by automating the calculations and providing the solutions. Here’s how to use one:

1. Input the Coefficients: Enter the values of ‘a’, ‘b’, and ‘c’ from the quadratic equation into the designated fields.
2. Click the “Solve” Button: Initiate the calculation process by pressing the “Solve” button.
3. Interpret the Results: The calculator will display the roots (solutions) of the quadratic equation.
4. Analyze the Solutions: Based on the discriminant value, interpret the nature of the solutions:
   • Two Real Solutions: If the discriminant is positive, there are two distinct real solutions.
   • One Repeated Real Solution: If the discriminant is zero, there is one repeated real solution.
   • Two Complex Solutions: If the discriminant is negative, there are two complex solutions.

Real-World Applications

Quadratic equations extend far beyond the realm of mathematics, finding practical applications in diverse fields:

1. Projectile Motion: The trajectory of a projectile, such as a ball or rocket, can be modeled using a quadratic equation, considering factors like initial velocity and gravitational pull.
2. Circuit Analysis: In electrical circuits, quadratic equations are used to calculate current, voltage, and power in complex circuits.
3. Optimization Problems: Quadratic equations calculator are employed in optimization problems, such as finding the maximum profit or minimum cost for a business.
4. Structural Design: Engineers utilize quadratic equations to analyze the stresses and strains in structures like bridges and buildings.

Additional Tips and Resources

• Visualize the Solutions: Graphing the quadratic equation can help visualize the solutions and their relationship to the discriminant.
• Practice with Examples: Engage in solving various quadratic equations to solidify your understanding and problem-solving skills.
• Utilize Online Resources: Explore interactive tutorials, practice problems, and online calculators to enhance your learning experience.

Conclusion

The quadratic formula calculator serves as a valuable tool for solving quadratic equations with ease and accuracy. By understanding the formula, its components, and its applications, you can confidently tackle quadratic problems in various academic and practical settings. Embrace the power of mathematics and unlock the secrets of quadratic equations!

Let’s break down the key components:

• a, b, and c: These are coefficients (numerical constants) in the quadratic equation (ax^2 + bx + c = 0). Notice that ‘a’ cannot be zero, or the equation wouldn’t be considered quadratic.
• b² – 4ac: This term is called the discriminant. Its value determines the nature of the solutions (roots) of the equation.
• ± : This symbol represents “plus or minus.” There will be two possible solutions for x in most cases.
• √ : This symbol represents the square root.

Understanding the Discriminant:

The discriminant plays a crucial role in determining the solutions of the quadratic formula. Here’s how:

• Discriminant > 0: This indicates two distinct real number solutions for x.
• Discriminant = 0: This results in one repeated real number solution for x.
• Discriminant < 0: This signifies two complex number solutions for x (involving imaginary unit ‘i’).

Quadratic Formula Calculator Using the Quadratic Formula Step-by-Step

Let’s solve an actual quadratic equation using the formula!

Example: A baseball player hits a home run, and the path of the ball can be modeled by the equation 2x^2 – 12x + 18 = 0 (where x represents the horizontal distance in meters). Find the two possible distances the ball travels before landing.

Step 1: Identify the coefficients.

• a = 2
• b = -12
• c = 18

Step 2: Calculate the discriminant.

• b² – 4ac = (-12)² – 4 * 2 * 18 = 0

Step 3: Apply the formula (since the discriminant is 0).

• x = (-b ± √(0)) / (2a)
• As the discriminant is 0, there will be one repeated solution. We can simplify by ignoring the ± part.
• x = (-(-12)) / (2 * 2) = 3

Therefore, the baseball travels a horizontal distance of 3 meters before landing (a repeated solution signifies a single peak in the trajectory).