Method

Solving a Quadratic Equation by Completing the Square

To solve a quadratic equation by completing the square, write the equation in the form

x^{2} + b x = c,

then complete the square of the expression on the left-hand side. Finally, the equation can be solved for

x

by taking square roots of both sides of the equation. To illustrate this, consider the following equation.

2 x^{2} + 12 x + 4 = 0

To solve the equation by completing the square, these five steps can be followed.

Write the Equation in the Form

x^{2} + b x = c

The Properties of Equality can be used jointly with inverse operations to rewrite the given equation in the form

x^{2} + b x = c .

2 x^{2} + 12 x + 4 = 0

SplitIntoFactors

Split into factors

2 x^{2} + 2 \cdot 6 x + 2 \cdot 2 = 0

FactorOut

Factor out $2$

2 (x^{2} + 6 x + 2) = 0

DivEqn

$LHS / 2 = RHS / 2$

x^{2} + 6 x + 2 = 0

SubEqn

$LHS - 2 = RHS - 2$

x^{2} + 6 x = - 2

If the equation is already in the form

x^{2} + b x = c,

this step is skipped.

Complete the Square on the Left-Hand Side of the Equation

To complete the square on the left-hand side of the equation, the square of one-half the coefficient of the

x -

term should be added to each side of the equation.

x^{2} + b x = c ⇓ x^{2} + b x + (\frac{b}{2})^{2} = c + (\frac{b}{2})^{2}

In the equation found previously,

b

is equal to

6 .

Therefore,

(\frac{6}{2})^{2}

should be added to both sides.

x^{2} + 6 x = - 2 ⇓ x^{2} + 6 x + (\frac{6}{2})^{2} = - 2 + (\frac{6}{2})^{2}

Next, the quotient can be simplified.

x^{2} + 6 x + 3^{2} = - 2 + 3^{2}

The expression on the left-hand side is now a perfect square trinomial.

Factor the Perfect Square Trinomial

Next, factor the perfect square trinomial.

x^{2} + 6 x + 3^{2} = - 2 + 3^{2}

SplitIntoFactors

Split into factors

x^{2} + 2 \cdot x \cdot 3 + 3^{2} = - 2 + 3^{2}

FacPosPerfectSquare

$a^{2} + 2 a b + b^{2} = (a + b)^{2}$

(x + 3)^{2} = - 2 + 3^{2}

Evaluate right-hand side

CalcPow

Calculate power

(x + 3)^{2} = - 2 + 9

AddTerms

Add terms

(x + 3)^{2} = 7

Take the Square Root of Both Sides of the Equation

The square root of both sides of the equation can now be taken to remove the exponent.

(x + 3)^{2} = 7

SqrtEqn

$LHS = RHS$

(x + 3)^{2} = 7

$a^{2} = \pm a$

x + 3 = \pm 7

Solve for

x

Finally, the resulting equations of the previous step need to be solved. These solutions will also be solutions to the original equation.

	$x + 3 = \pm 7$
Write as two equations	$x + 3 = 7$	$x + 3 = - 7$
Solve for $x$	$x = - 3 + 7$	$x = - 3 - 7$

Therefore, the solutions of the given equation are $x = - 3 + 7$ and $x = - 3 - 7 .$

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Method

Solving a Quadratic Equation by Completing the Square

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