Completing the Square Calculator

Completing the Square Calculator is an online tool that helps to complete the square of a quadratic equation and calculate its roots. We can convert the quadratic expression to vertex form by using completing the squares method and then solve it.

What is Completing the Square Calculator?

Completing the Square Calculator helps to compute the roots of a given quadratic equation by completing its square. Algebraic equations that have only 1 variable and are of the second degree are known as quadratic equations. To use completing the square calculator, enter the values in the input boxes.

Completing the Square Calculator

How to Use Completing the Square Calculator?

Please follow the steps below to find the roots of a quadratic equation by completing its square using completing the square calculator:

• Step 1: Go to Cuemath's online completing the square calculator.
• Step 2: Enter the values in the given input boxes of completing the square calculator.
• Step 3: Click on the "Solve" button to calculate the roots of the given quadratic equation by completing its square.
• Step 4: Click on "Reset" to clear the fields and enter new values.

How Does Completing the Square Calculator Work?

We can apply 4 methods to determine the roots of a quadratic equation. These are, completing the squares of the quadratic equation, factorizing the quadratic equation, applying the quadratic formula, and using the graphing method. The steps to complete the square of a quadratic equation are given as follows:

• Suppose we have a quadratic equation expressed as ax² + bx + c = 0.
• We need to ensure that the coefficient of x is always 1. Thus we divide the entire quadratic equation by the coefficient of x; x² + bx/a -c/a = 0.
• Keeping the variable terms on one side and the constants on the other we get, x² + bx/a = -c/a.
• We add the term (b/2a)² on both sides; x² + bx/a + (b/2a)² = -c/a + (b/2a)²
• This helps us to express the L.H.S as a perfect square; (x + b/2a)² = -c/a + (b/2a)².
• The vertex form is given as a(x + b/2a)² + (c - b²/4a) = 0. (We multiply the entire equation by a).
• We can now solve this equation to determine the roots.

Thus, ax² + bx + c can be expressed as a(x + d)² + e by completing the square.

Here d = b / 2a and e = c - (b²/4a).

Solved Examples on Completing the Square

Example 1: Solve quadratic equation x² + 6x + 7 = 0 using completing the square method and verify it using the online completing the square calculator.

Solution:

Given: a = 1, b = 6, c = 7

Converting quadratic expression in the form ax² + bx + c = 0 to the vertex form a(x+d)² + e = 0, where d = b/2a, and e = c - b²/(4a)

d = b/2a = 6 / 2 = 3

e = c - b²/(4a) = 7 - 6² / 4 = 7 - 9 = -2

Substitute the d = 3 and e = -2 in the vertex form a(x+d)² + e

1(x + 3)² + (-2) = 0

(x + 3)² = 2

Taking the square root on both sides

x + 3 = +√2

x = √2 - 3, -√2 - 3

x = -1.59 , -4.41

Example 2: Solve quadratic equation 8x² + 20x - 3 = 0 using the completing square method and verify it using the online completing the square calculator.

Solution:

Given: a = 8, b = 20, c = -3

Converting quadratic expression in the form ax² + bx + c = 0 to the vertex form a(x+d)² + e = 0, where d = b/2a, and e = c - b²/(4a)

d = b/2a = 20 / 16 = 5/4

e = c - b²/(4a) = -3 - 20² / 32 = -15.5

Substitute the d = 1/8 and e = -3.13 in the vertex form a(x+d)² + e

8(x + 5/4)² + (-15.5) = 0

(x + 5/4)² = 15.5 / 8 = 1.937

Taking the square root on both sides

(x + 5/4) = +1.39

x = 1.39 - 5/4, -1.39 - 5/4

x = 0.14 , -2.64

Similarly, you can try completing the square calculator to find the roots of the quadratic equation using completing the square method for:

• 7x² - 10x + 16 = 0
• x² - 5x + 6 = 0

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