Question

Find the discriminant of the quadratic equation 16x^2-72x+81=0 and hence find the nature of its roots​

Answer 1

Answer:

16x^2-72x+81=0

STEP 1

:

Equation at the end of step 1

(24x2 - 72x) + 81 = 0

STEP

2

:

Trying to factor by splitting the middle term

2.1 Factoring 16x2-72x+81

The first term is, 16x2 its coefficient is 16 .

The middle term is, -72x its coefficient is -72 .

The last term, "the constant", is +81

Step-1 : Multiply the coefficient of the first term by the constant 16 • 81 = 1296

Step-2 : Find two factors of 1296 whose sum equals the coefficient of the middle term, which is -72 .

-1296 + -1 = -1297

-648 + -2 = -650

-432 + -3 = -435

-324 + -4 = -328

-216 + -6 = -222

-162 + -8 = -170

-144 + -9 = -153

-108 + -12 = -120

-81 + -16 = -97

-72 + -18 = -90

-54 + -24 = -78

-48 + -27 = -75

-36 + -36 = -72 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -36 and -36

16x2 - 36x - 36x - 81

Step-4 : Add up the first 2 terms, pulling out like factors :

4x • (4x-9)

Add up the last 2 terms, pulling out common factors :

9 • (4x-9)

Step-5 : Add up the four terms of step 4 :

(4x-9) • (4x-9)

Which is the desired factorization

Multiplying Exponential Expressions:

2.2 Multiply (4x-9) by (4x-9)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is (4x-9) and the exponents are :

1 , as (4x-9) is the same number as (4x-9)1

and 1 , as (4x-9) is the same number as (4x-9)1

The product is therefore, (4x-9)(1+1) = (4x-9)2

Equation at the end of step

2

:

(4x - 9)2 = 0

STEP

3

:

Solving a Single Variable Equation:

3.1 Solve : (4x-9)2 = 0

(4x-9) 2 represents, in effect, a product of 2 terms which is equal to zero

For the product to be zero, at least one of these terms must be zero. Since all these terms are equal to each other, it actually means : 4x-9 = 0

Add 9 to both sides of the equation :

4x = 9

Divide both sides of the equation by 4:

x = 9/4 = 2.250

Step-by-step explanation:

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