menu_book {{ printedBook.name}}

arrow_left {{ state.menu.current.label }}

arrow_left {{ state.menu.current.current.label }}

arrow_left {{ state.menu.current.current.current.label }}

{{ result.displayTitle }} *navigate_next*

{{ result.subject.displayTitle }}

{{ 'math-wiki-no-results' | message }}

{{ 'math-wiki-keyword-three-characters' | message }}

{{ r.avatar.letter }}

{{ r.name }} {{ r.lastMessage.message.replace('TTREPLYTT','') }} *people*{{keys(r.currentState.members).length}} *schedule*{{r.lastMessage.eventTime}}

{{ r.getUnreadNotificationCount('total') }} +

{{ u.avatar.letter }}

{{ u.displayName }} (you) {{ r.lastMessage.message.replace('TTREPLYTT','') }} *people*{{keys(r.currentState.members).length}} *schedule*{{r.lastMessage.eventTime}}

{{ r.getUnreadNotificationCount('total') }} +

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. *expand_more*
*expand_more*
*expand_more*
*expand_more*
*expand_more*

1

Factor Out the Coefficient of x2

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

Identify the Constant Needed to Complete the Square

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

Complete the Square

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

$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. 4

Factor the Perfect Square Trinomial

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.

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

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

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

5

Simplify the Equation

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.

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

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 $

The solutions of a quadratic equation can be interpreted graphically as the zeros of the related quadratic function. Therefore, the number of solutions of a quadratic equation is the same as the number of zeros of the related function. *two* zeros, the equation ax2+bx+c=0 has *two* solutions. Similarly, if the function has *one* zero, the equation has *one* solution. Finally, if the function does not have *any* zeros, the equation has *no* real solutions.

$Quadratic Equationax_{2}+bx+c=0 Quadratic Functiony=ax_{2}+bx+c $

If the function has 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.

2x2−4x+10=0

Show Solution *expand_more*

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

{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

{{ focusmode.exercise.exerciseName }}

close

Community rate_review

{{ r.avatar.letter }}

{{ u.avatar.letter }}

+