## HOW TO FIND ZEROS OF QUADRATIC POLYNOMIAL

On the webpage,"how to find zeros of quadratic polynomial" we are going to learn the methods of solving quadratic equation.

## Methods of solving quadratic equations

Actually we have three methods to solve a quadratic equations

## Find zeroes by factoring method

In a quadratic "Leading coefficient" means "coefficient of x²".

(i) If the coefficient is 1 we have to take the constant term and we have to split it as two parts.

(ii) The product of two parts must be equal to the constant term and the simplified value must be equal to the middle term (or) x term.

(iii) Now we have to write these numbers in the form of (x + a) and (x +b)

(iv) By equating the factors equal to zero.We can find the value of x.  solving quadratic equations by factoring Let us see example problem on "how to find zeros of quadratic polynomial".

Example 1 :

Find the zeros of the quadratic equation x² + 17 x + 60 by factoring.

Solution : Since it is 1. We are going to take the last number. That is 60 and we are going to find factors of 60.

All terms are having positive sign. So we have to put positive sign for both factors. Here,

10  x 6 = 60 but 10 + 6 = 16 not 17

15 x 4 = 60 but 15 + 4 = 19 not 17

12 x 5 = 60 and 12 + 5 = 17

2 x 30 = 60 but 2 + 30 = 32 not 17 (x + 12) (x + 5) are the factors

x + 12 = 0        x + 5 = 0

x = -12            x = -5

Hence, zeros of the given quadratic equation are -12 and -5

This is just example of solving quadratic equations by factoring.If you need more example problems of solving quadratic equations by factoring please click the below link.

Need more examples

Factoring quadratic equations when a isn't 1

(i) If it is not 1 then we have to multiply the coefficient of x² by the constant term and we have to split it as two parts.

(ii) The product of two parts must be equal to the constant term and the simplified value must be equal to the middle term (or) x term.

(iii) Divide the factors by the coefficient of x². Simplify the factors by the coefficient of x² as much as possible.

(iv) Write the remaining number along with x.

Example 2 :

Find the zeros of the quadratic equation 2 x² + x - 6 by factoring.

Solution : We get -12, now we have to split -12 as the multiple of two numbers. Since the last term is having negative sign, we have to put negative sign for the least number. Now we have to divide the two numbers 4 and -3 by the coefficient of x² that is 2. If it is possible we can simplify otherwise we have to write the numbers along with x. (x + 2) (2x - 3)  = 0

x + 2 = 0         2 x - 3 = 0

x = -2             2 x = 3

x = 3/2

Hence, the zeros of the given quadratic equation are -2 and 3/2.

This is just one example problem to show solving quadratic equations by factoring. If you want more example please click the below link.

Need more examples

Let us see the next concept on "how to find zeros of quadratic polynomial".

## Find zeros of quadratic equation by using formula

(i) First we have to compare the given quadratic equation with the general form of quadratic equation ax² + bx + c = 0

(ii) Here the coefficient of x² is a, coefficient of x is b and the constant term is c.

(iii) Then we have to apply those values in the formula -b
± √ (b² - 4 a c)/2a

Let us see an example problem on "how to find zeros of quadratic polynomial".

Example 3 :

Find zeros of the quadratic equation x²- 7x + 12 by using formula

Solution :

a = 1     b = -7    c = 12 = 8/2, 6/2

= 4, 3

Hence, the zeros of the given quadratic equation are 4 and 3.

If you need more examples please visit the page "Solving quadratic equation by using formula".

Let us the next concept on "how to find zeros of quadratic polynomial".

## Find zeros of quadratic equation by completing the square method

(i) First we have to check whether the coefficient of x² is 1 or not. If yes we can follow the second step. Otherwise we have to divide the entire equation by the coefficient of x².

(ii) Bring the constant term which we find on the left side to the right side.

(iii) We have to add the square of half of the coefficient of "x" on both sides.

(iv) Now the three terms on the left side will be in the form of a² + 2 a b + b² (or) a² - 2 ab + b².

(v) Then, we can write (a + b)² for a² + 2 a b + b² and       (a- b)² for a² - 2 a b + b². Then we have to solve for x by simplification.

Let us see an example problem on "how to find zeros of quadratic polynomial".

Example 4 :

Find the zeros of the quadratic equation  x²+6x-7  by completing the square method

Solution : how to find zeros of quadratic polynomial (x + 3)² = 16

x + 3 = √ 16

x + 3 = ± 4

x + 3 = 4             x + 3 = - 4

x = 4 - 3                   x = - 4 - 3

x = 1                       x = - 7

Hence, the zeros of the quadratic equation are 1 and -7. WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6