ShadowWolfie
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Solve the equation with special factors.
x^2 - 81 = 0

Sagot :

Answer:

x = ± 9

Step-by-step explanation:

x² - 81 = 0 ← is a difference of squares and factors in general as

a² - b² = (a - b)(a + b) , then

x² - 81 = 0

x² - 9² = 0

(x - 9)(x + 9) = 0

Equate each factor to zero and solve for x

x - 9 = 0 ⇒ x = 9

x + 9 = 0 ⇒ x = - 9

ᴜᴋɪʏᴏ

Hello ShadowWolfie!

[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]

Solve the equation with special factors.

x² - 81 = 0.

[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]

You can solve this question using 3 methods :-

  1. By using the quadratic formula.
  2. By finding the square root.
  3. By using the difference of the squares.

Let's take a look at all the 3 methods.

__________________

1. By Using Quadratic Formula :-

[tex]x ^ { 2 } - 81 = 0[/tex]

We can use the biquadratic formula straight away as it is already in the form of ax² + bx + c. Now, we know that the biquadratic formula is :- [tex]\frac{ - b \: ± \: \sqrt{ {b}^{2} - 4ac} }{2a}[/tex]. In this equation..

  • a = x
  • b = 0
  • c = -81

Now, substitute these values in the formula..we get..

[tex] \\ x=\frac{0±\sqrt{0^{2}-4\left(-81\right)}}{2} [/tex]

Let's simplify it.

[tex]x=\frac{0±\sqrt{0^{2}-4\left(-81\right)}}{2} \\ x=\frac{0±\sqrt{-4\left(-81\right)}}{2} \\ x=\frac{0±\sqrt{324}}{2} \\ x=\frac{0±18}{2} \\ \boxed{\boxed{ \bf \: x = ±9}}[/tex]

__________________

2. By Finding The Square Root :-

[tex]x ^ { 2 } - 81 = 0[/tex]

Add zero to both the sides..our equation will become..

[tex]x^{2}=81 [/tex]

Now, take the square root on both the sides.

[tex]x^{2}=81 \\ \sqrt{ {x}^{2} } = \sqrt{81} \\ \boxed{ \boxed{\bf x = ±9}}[/tex]

__________________

3. By Using The Difference of The Squares :-

[tex]x ^ { 2 } - 81 = 0[/tex]

Let's rewrite this equation as :- x² - 9². Now let's solve it using the algebraic identity ⇨ a² - b² = (a + b) (a - b). So..

[tex]x ^ { 2 } - 81 = 0 \\ {x}^{2} - {9}^{2} = 0 \\ \left(x-9\right)\left(x+9\right)=0 [/tex]

Let's solve for each term..

[tex](x - 9) = 0 \\ \underline{ \underline{x = + 9}}[/tex]

And..

[tex](x + 9) = 0 \\ \underline{ \underline{x = - 9}}[/tex]

So,

[tex]\boxed{ \boxed{\bf x = ±9}}[/tex]

__________________

  • By using the 3 different methods (you can choose whichever method you find is easier), we got the answer as x = ± 9 .

__________________

Hope it'll help you!

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