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Sometimes, quadratic equations have no solutions that can be expressed using real numbers. However, it is possible to use complex numbers to express these non-real solutions.

To solve a quadratic equation written in standard form ax2+bx+c=0, the Quadratic Formula can be used.

In this formula, the discriminant b2−4ac determines the number of real solutions of the quadratic equation.

The Quadratic Formula can be derived by completing the square given the standard form of the quadratic equation ax2+bx+c=0. This method will be used to isolate the x-variable. To complete the square, there are five steps to follow. ### 1

It is easier to complete the square when the quadratic expression is written in the form x2+bx+c. Therefore, the coefficient a should be factored out.
Since the equation is quadratic, the coefficient a is not equal to 0. Therefore, both sides of the equation can be divided by a.
### 2

The next step is to rewrite the equation by moving the existing constant to the right-hand side. To do so, $ac $ will be subtracted from both sides of the equation.
The constant needed to complete the square can now be identified by focusing on the x-term, while ignoring the rest. One way to find this constant is by squaring half the coefficient of the x-term, which in this case is $ab .$
Note that leaving the constant as a power makes the next steps easier to perform.
### 3

The square can now be completed by adding the constant found in Step 2 to both sides of the equation.
### 4

The perfect square trinomial can now be factored and rewritten as the square of a binomial.
The process of completing the square is now finished.
### 5

Finally, the right-hand side of the equation can be simplified.
Now, there is only one x-term. To isolate x, it is necessary to take square roots on both sides of the equation. This results in both a positive and a negative term on the right-hand side.
Now, the equation can be further simplified to isolate x.
Finally, the Quadratic Formula has been obtained.

Factor Out the Coefficient of x2

Identify the Constant Needed to Complete the Square

Complete the Square

$x_{2}+ab x=-ac ⇕Perfect Square Trinomialx_{2}+ab x+(2ab )_{2} =Constant-ac +(2ab )_{2} $

The first three terms form a perfect square trinomial, which can be factored as the square of a binomial. The other two terms do not contain the variable x. Therefore, their value is constant.
Factor the Perfect Square Trinomial

$x_{2}+ab x+(2ab )_{2}=-ac +(2ab )_{2}$

DenomMultFracToNumber

$2⋅2a =a$

$x_{2}+2(2ab )x+(2ab )_{2}=-ac +(2ab )_{2}$

CommutativePropMult

Commutative Property of Multiplication

$x_{2}+2x(2ab )+(2ab )_{2}=-ac +(2ab )_{2}$

FacPosPerfectSquare

a2+2ab+b2=(a+b)2

$(x+2ab )_{2}=-ac +(2ab )_{2}$

Simplify the Equation

$(x+2ab )_{2}=-ac +(2ab )_{2}$

Simplify right-hand side

CommutativePropAdd

Commutative Property of Addition

$(x+2ab )_{2}=(2ab )_{2}−ac $

PowQuot

$(ba )_{m}=b_{m}a_{m} $

$(x+2ab )_{2}=(2a)_{2}b_{2} −ac $

PowProdII

$(ab)_{m}=a_{m}b_{m}$

$(x+2ab )_{2}=4a_{2}b_{2} −ac $

ExpandFrac

$ba =b⋅4aa⋅4a $

$(x+2ab )_{2}=4a_{2}b_{2} −a⋅4ac⋅4a $

CommutativePropMult

Commutative Property of Multiplication

$(x+2ab )_{2}=4a_{2}b_{2} −4a⋅a4ac $

ProdToPowTwoFac

a⋅a=a2

$(x+2ab )_{2}=4a_{2}b_{2} −4a_{2}4ac $

SubFrac

Subtract fractions

$(x+2ab )_{2}=4a_{2}b_{2}−4ac $

$x+2ab =±4a_{2}b_{2}−4ac $

Solve for x

SqrtQuot

$ba =b a $

$x+2ab =±4a_{2} b_{2}−4ac $

SqrtProd

$a⋅b =a ⋅b $

$x+2ab =±2a_{2} b_{2}−4ac $

SqrtPowToNumber

$a_{2} =a$

$x+2ab =±2ab_{2}−4ac $

SubEqn

$LHS−2ab =RHS−2ab $

$x=-2ab ±2ab_{2}−4ac $

MoveNegFracToNum

Put minus sign in numerator

$x=2a-b ±2ab_{2}−4ac $

AddSubFrac

Add and subtract fractions

$x=2a-b±b_{2}−4ac $

In the Quadratic Formula, the expression b2−4ac, which is under the radical symbol, is called the discriminant.

A quadratic equation can have two, one, or no real solutions. Since the discriminant is under the radical symbol, its value determines the number of real solutions of a quadratic equation.

Value of the Discriminant | Number of Real Solutions |
---|---|

b2−4ac>0 | 2 |

b2−4ac=0 | 1 |

b2−4ac<0 | 0 |

Moreover, the discriminant determines the number of x-intercepts of the graph of the related quadratic function.

When the discriminant is negative, there are no real solutions of the quadratic equation. However, the square root of a negative number can be written as an imaginary number. This way, complex solutions of a quadratic equation can be found.

Solve the equation using the quadratic formula.

2x2−4x+10=0

Show Solution

Since the quadratic equation is given in standard form, we can immediately identify the constants
Notice that there is a negative number inside the square root. Thus, the equation has no real solutions. However, we can calculate the complex roots using the identity
The solutions to the equation are x=1+2i and x=1−2i.

$a=2,b=-4,andc=10.$

Let's substitute these values into the quadratic formula to solve the equation. 2x2−4x+10=0

UseQuadForm

Use the Quadratic Formula: a=2,b=-4,c=10

$x=2⋅2-(-4)±(-4)_{2}−4⋅2⋅10 $

NegNeg

-(-a)=a

$x=2⋅24±(-4)_{2}−4⋅2⋅10 $

CalcPowProd

Calculate power and product

$x=44±16−80 $

SubTerm

Subtract term

$x=44±-64 $

$x=44±-64 $

SqrtNegToSqrtI

$-a =a ⋅i$

$x=44±64 ⋅i $

CalcRoot

Calculate root

$x=44±8i $

SimpQuot

Simplify quotient

x=1±2i

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