# Completing The Square In Quadratics Calculator

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The Completing The Square In Quadratics Calculator will calculate and:

1. Complete the square in a quadratic equation of the form ax2 + bx + c = 0.

Completing The Square In Quadratics Calculator Parameters: The quadratic equation is assumed to have two distinct roots.

 🖹 Normal View🗖 Full Page View Calculator Precision (Decimal Places)0123456789101112131415 Coefficient a (a) Coefficient b (b) Constant c (c)
Completing the Square in Quadratics Formula and Calculations Writing a quadratic equation in the form that shows the completing of the square:(x )2 + = 0 a(x + b/2a)2 + (c - b2/4a2) = 0(x + /2 × )2 + ( - 2/4 × 2) = 0(x + /)2 + ( - /4 × ) = 0(x + )2 + (/ - /) = 0(x + )2 + (/) = 0(x )2 + = 0 Coefficient a (a) = Coefficient b (b) = Constant c (c) =

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## Theoretical description

A quadratic equation is a second-order equation with one variable of the form

y = ax2 + bx + c = 0

where x is the variable, a and b are coefficients and c is a constant.

For example, the equation 3x2 - 4x + 1 = 0 is a quadratic equation, where a = 3, b = -4 and c = 1.

A quadratic equation may have one or two roots or it may not have any root. One of the methods used for solving quadratic equations when they have two roots consists of completing the square. We must therefore try to express a given quadratic equation

ax2 + bx + c = 0

in the form

a(x+p)2 + q = 0

where p and q are numbers.

Let's write p and q in terms of the (known) coefficients a and b and the constant c. We have

ax2 + bx + c = a(x + p)2 + q
ax2 + bx + c = a(x2 + 2px + p2) + q
ax2 + bx + c = ax2 + 2apx + ap2 + q

Comparing the like terms on both sides yields

b = 2ap

Hence,

p = b/2a

and

c = ap2 + q

Thus,

q = c - ap2
= c - a ∙ (b/2a)2
= c - b2/4a2

Hence, we complete the square by writing the quadratic equation as

a(x + b/2a)2 + (c - b2/4a2) = 0

For example, we can write the quadratic equation 3x2 - 4x + 1 = 0 (a = 3, b = -4 and c = 1) as

3(x + -4/2∙3)2 + (1 - -42/4 ∙ 32 ) = 0
3(x + -4/6)2 + (1 - 16/36) = 0
3(x - 2/3)2 + (36/36 - 16/36) = 0
3(x - 2/3)2 + 20/36 = 0
3(x - 2/3)2 + 5/9 = 0

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