Solving Quadratic Equations Online

A Quadratic equation is an equation with the highest degree two. The standard form is written as ax2+bx+c=0 where a is not equal to zero and a,b are the coefficients of x2and x respectively; c is a constant. Some of the examples are, 2x2+3x+5; x2-4x+5; 4x2-16 etc. Let us now learn how to solve Quadratic Equations. The following are the different methods used in solving Quadratic Equations,

  • Factoring method

  • Completing the square method

  • Using the Quadratic Formula

Any equation of the form ax2+bx+c=0 and ax2+bx=0 where c is zero can be solved using the factoring method by finding the factors of the equation and solving for x.

 

Having problem with Quadratic Formula Problems keep reading my upcoming posts, i will try to help you.

 

Examples On Solving Quadartic Equations


For example, solve x2+4x-5=0 for x

Solution: Given equation x2+4x-5=0

To factor the equation we find the product of the coefficient of x2 and the constant term which is,

Product= (1)(-5)=-5

List out the factors of -5 and choose the appropriate factors such that their sum would be equal to the coefficient of x which is 4

Factors of- 5 are -1 & 5; the sum (5-1) gives -4 and the product (5)(-1)= -5

Now the middle term 4x is split into the factors as -x and 5x

x2-x+5x-5x

Re-grouping the terms, x(x-1)+ 5(x-1)

Taking (x-1) common gives (x-1)(x+5)

Solving for x, x-1=0; x+5=0

Solutions are x=1 and x=-5

 

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Let us solve Quadratic Equations using completing the square method. For example, solve x2+6x-7=0

Solution: Given equation x2+6x-7=0

The constant term is taken to the other side of the equal to sign

x2+6x = 7

Now consider the coefficient of the x term which is 6, half the value and square it which gives 6/2=32=9

Converting the left hand side to a squared form by adding 9 on both sides

x2+6x+9= 7+9

(x+3)2=16 [a2+2ab+b2=(a+b)2]

Square root on both sides

(x+3) = 4

We get, x+3=4 and x+3=-4

Solutions are x = 1 and x =-7

 

Sometimes the quadratic equations cannot be factored, in such cases the Quadratic formula is used to solve the equations. For the standard form of the equation ax2+bx+c, the solutions using the formula is given as, [-bsqrt(b2-4ac)]/2a.

Here the given equation is compared to the standard form, the values substituted appropriately and finally solved for the variable

For example, solve 3x2+5x+1=0 for x

Solution: Given equation 3x2+5x+1=0

Comparing the terms with the standard equation ax2+bx+c=0

a=3, b=5 and c=1

Substituting the values of a,b and c in the formula and simplifying gives

x= [-bsqrt(b2-4ac)]/2a

x= [[-5sqrt(52-4(3)(1))]/2(3)

x= [-5sqrt(13)]/6

Solutions are, x= [-5+sqrt(13)]/6 and x= [-5 - sqrt(13)]/6

 

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