Completing Square Calculator

Completing the Square Calculator

Enter the coefficients of your quadratic equation (ax2 + bx + c = 0) below.

Results:

Enter values and click 'Calculate' to see the steps and solutions.

// Helper for Greatest Common Divisor (GCD) function gcd(a, b) { return b === 0 ? a : gcd(b, a % b); } // Helper for formatting fractions function formatFraction(numerator, denominator) { if (denominator === 0) return "Undefined"; if (numerator === 0) return "0"; if (denominator === 1) return numerator.toString(); var common = gcd(Math.abs(numerator), Math.abs(denominator)); var num = numerator / common; var den = denominator / common; if (den < 0) { // Ensure denominator is positive num = -num; den = -den; } if (num % den === 0) { // If it's an integer return (num / den).toString(); } return num + "/" + den; } // Helper for formatting square roots function formatSqrt(value) { if (value === 0) return "0"; if (value < 0) return "√(" + value + ")"; // Should be handled by complex numbers var sqrtVal = Math.sqrt(value); if (sqrtVal === Math.floor(sqrtVal)) { return sqrtVal.toString(); } return "√" + value; } function calculateCompletingSquare() { var a = parseFloat(document.getElementById("coeffA").value); var b = parseFloat(document.getElementById("coeffB").value); var c = parseFloat(document.getElementById("coeffC").value); var resultDiv = document.getElementById("calculatorResult"); var output = '

Results:

'; if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = output + 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { resultDiv.innerHTML = output + 'Coefficient \'a\' cannot be zero for a quadratic equation.'; return; } output += 'Given quadratic equation: ' + a + 'x2 + ' + b + 'x + ' + c + ' = 0'; output += '

Steps to Complete the Square:

    '; // Step 1: Divide by 'a' if a != 1 var b_prime = b; var c_prime = c; var step1_equation = "; if (a !== 1) { b_prime = b / a; c_prime = c / a; step1_equation = 'x2 + ' + formatFraction(b, a) + 'x + ' + formatFraction(c, a) + ' = 0'; output += '
  1. Divide the entire equation by ' + a + ': ' + step1_equation + '
    2. '; } else { step1_equation = 'x2 + ' + b + 'x + ' + c + ' = 0'; output += '
  2. (Coefficient \'a\' is 1, no division needed): ' + step1_equation + '
    3. '; } // Step 2: Move constant to the right side var right_side_step2 = -c_prime; output += '
  3. Move the constant term to the right side of the equation: x2 + ' + formatFraction(b_prime * a, a) + 'x = ' + formatFraction(-c_prime * a, a) + '
    4. '; // Step 3: Find (b'/2)^2 and add to both sides var half_b_prime = b_prime / 2; var square_half_b_prime = half_b_prime * half_b_prime; output += '
  4. Take half of the coefficient of x (' + formatFraction(b_prime * a, a) + '), square it, and add to both sides: '; output += 'Half of ' + formatFraction(b_prime * a, a) + ' is ' + formatFraction(b_prime * a, 2 * a) + '. Squared is (' + formatFraction(b_prime * a, 2 * a) + ')2 = ' + formatFraction(b_prime * b_prime * a * a, 4 * a * a) + "; output += 'x2 + ' + formatFraction(b_prime * a, a) + 'x + ' + formatFraction(b_prime * b_prime * a * a, 4 * a * a) + ' = ' + formatFraction(-c_prime * a, a) + ' + ' + formatFraction(b_prime * b_prime * a * a, 4 * a * a) + '
    5. '; // Step 4: Factor the left side and simplify the right side var right_side_simplified_num = -c_prime * 4 * a * a + b_prime * b_prime * a * a; var right_side_simplified_den = 4 * a * a; var right_side_simplified_fraction = formatFraction(right_side_simplified_num, right_side_simplified_den); output += '
  5. Factor the left side as a perfect square and simplify the right side: '; output += '(x + ' + formatFraction(b_prime * a, 2 * a) + ')2 = ' + right_side_simplified_fraction + '
    6. '; // Step 5: Take the square root of both sides output += '
  6. Take the square root of both sides: '; output += 'x + ' + formatFraction(b_prime * a, 2 * a) + ' = ±' + formatSqrt(right_side_simplified_num / right_side_simplified_den) + '
    7. '; // Step 6: Solve for x (roots) output += '
  7. Solve for x: '; var discriminant = b * b – 4 * a * c; var x1_display, x2_display; if (discriminant >= 0) { var sqrt_discriminant = Math.sqrt(discriminant); var x1 = (-b + sqrt_discriminant) / (2 * a); var x2 = (-b – sqrt_discriminant) / (2 * a); x1_display = formatFraction(-b * 2 * a + sqrt_discriminant * 2 * a, 4 * a * a); // Attempt to keep fraction form x2_display = formatFraction(-b * 2 * a – sqrt_discriminant * 2 * a, 4 * a * a); if (sqrt_discriminant === Math.floor(sqrt_discriminant)) { // If discriminant is a perfect square x1_display = formatFraction(-b + sqrt_discriminant, 2 * a); x2_display = formatFraction(-b – sqrt_discriminant, 2 * a); } else { // Use decimals if sqrt is not an integer x1_display = x1.toFixed(4); x2_display = x2.toFixed(4); } output += 'x = ' + formatFraction(-b * 2 * a, 4 * a * a) + ' ± ' + formatSqrt(discriminant) + '/' + formatFraction(2 * a * 2 * a, 2 * a) + ''; output += 'x1 = ' + x1_display + ''; output += 'x2 = ' + x2_display + '
    8. '; } else { var abs_discriminant = Math.abs(discriminant); var sqrt_abs_discriminant = Math.sqrt(abs_discriminant); var real_part = formatFraction(-b, 2 * a); var imag_part_num = formatSqrt(abs_discriminant); var imag_part_den = formatFraction(2 * a * 2 * a, 2 * a); output += 'x = ' + real_part + ' ± ' + imag_part_num + 'i / ' + imag_part_den + ''; output += 'x1 = ' + real_part + ' + ' + formatFraction(sqrt_abs_discriminant, 2 * a) + 'i'; output += 'x2 = ' + real_part + ' - ' + formatFraction(sqrt_abs_discriminant, 2 * a) + 'i'; } output += '
'; // Vertex Form var h_num = -b; var h_den = 2 * a; var k_num = 4 * a * c – b * b; var k_den = 4 * a; output += '

Vertex Form (y = a(x – h)2 + k):

'; output += 'The vertex form of the equation is: y = ' + a + '(x + ' + formatFraction(h_num, h_den) + ')2 + ' + formatFraction(k_num, k_den) + ''; output += 'The vertex (h, k) is: (' + formatFraction(h_num, h_den) + ', ' + formatFraction(k_num, k_den) + ')'; resultDiv.innerHTML = output; }

Understanding the Completing the Square Method

Completing the square is a powerful algebraic technique used to solve quadratic equations, convert quadratic functions into vertex form, and derive the quadratic formula itself. It transforms a standard quadratic equation (ax2 + bx + c = 0) into a perfect square trinomial, making it easier to isolate the variable x.

What is a Perfect Square Trinomial?

A perfect square trinomial is a trinomial that results from squaring a binomial. For example:

  • (x + k)2 = x2 + 2kx + k2
  • (x – k)2 = x2 – 2kx + k2

The key characteristic is that the constant term (k2) is the square of half the coefficient of the x-term (2k).

Why Use Completing the Square?

  1. Solving Quadratic Equations: It provides a systematic way to find the roots (solutions) of any quadratic equation, even those that cannot be easily factored.
  2. Finding the Vertex of a Parabola: When a quadratic function is in vertex form, y = a(x – h)2 + k, the vertex of its parabolic graph is (h, k). Completing the square helps convert the standard form to the vertex form.
  3. Deriving the Quadratic Formula: The quadratic formula itself is derived by applying the completing the square method to the general quadratic equation ax2 + bx + c = 0.

Step-by-Step Guide to Completing the Square

Let's walk through the process using the general quadratic equation: ax2 + bx + c = 0

  1. Ensure the Leading Coefficient (a) is 1: If 'a' is not 1, divide the entire equation by 'a'. This will give you an equation in the form x2 + (b/a)x + (c/a) = 0.
  2. Move the Constant Term: Isolate the x2 and x terms on one side of the equation by moving the constant term (c/a) to the right side. The equation becomes x2 + (b/a)x = -(c/a).
  3. Find the Term to Complete the Square: Take half of the coefficient of the x-term (which is b/a), and then square it. This value is ((b/a)/2)2 = (b/2a)2 = b2/(4a2).
  4. Add the Term to Both Sides: Add the value calculated in step 3 to both sides of the equation. This ensures the equation remains balanced and the left side becomes a perfect square trinomial: x2 + (b/a)x + b2/(4a2) = -(c/a) + b2/(4a2).
  5. Factor the Perfect Square Trinomial: The left side can now be factored as a squared binomial: (x + b/2a)2. Simplify the right side by finding a common denominator.
  6. Take the Square Root of Both Sides: Apply the square root to both sides of the equation. Remember to include both the positive and negative roots (±). This will give you: x + b/2a = ±√[simplified right side].
  7. Solve for x: Isolate x by subtracting b/2a from both sides. This will give you the two solutions (roots) for the quadratic equation.

Example: Solving x2 + 6x + 5 = 0

  1. a = 1: The leading coefficient is already 1. x2 + 6x + 5 = 0
  2. Move Constant: x2 + 6x = -5
  3. Find Term: Half of 6 is 3. 32 = 9.
  4. Add Term: x2 + 6x + 9 = -5 + 9
  5. Factor and Simplify: (x + 3)2 = 4
  6. Take Square Root: x + 3 = ±√4 x + 3 = ±2
  7. Solve for x: x = -3 ± 2 x1 = -3 + 2 = -1 x2 = -3 - 2 = -5

The solutions are x = -1 and x = -5.

Using the Completing the Square Calculator

Our calculator simplifies this process for you. Simply input the coefficients 'a', 'b', and 'c' from your quadratic equation (ax2 + bx + c = 0) into the respective fields. Click 'Calculate', and the tool will instantly display the step-by-step solution using the completing the square method, along with the final roots and the vertex form of the equation.

This tool is perfect for students, educators, and anyone needing to quickly solve quadratic equations or understand the mechanics of completing the square.

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