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

Actually we have three methods to solve a quadratic equations

- Solving quadratic equation by factoring
- By using quadratic formula
- By using completing the square 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.

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.

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

(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".

(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.

- Solving quadratic equation by completing the square
- Solving quadratic equation by sum and product of roots
- Solving quadratic equation by formula
- Solving quadratic inequalities graphically
- Solving quadratic inequalities algebraically
- Solving word problems using quadratic equation

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