Question

Square root of 6x-12=x-2

Answer 1

x = 8 and x = 2

Explanation:

Given equation:

`√(6x-12)=x-2`

To solve the given equation, we can use algebraic operations to simplify the equation.

Begin by squaring both sides of the equation:

`(√(6x-12))^2=(x-2)^2`

Simplify:

`6x-12=x^2-4x+4`

Subtract 6x from both sides of the equation:

`6x-12-6x=x^2-4x+4-6x`

`-12=x^2-10x+4`

Add 12 to both sides of the equation:

`-12+12=x^2-10x+4+12`

`0=x^2-10x+16`

`x^2-10x+16=0`

Now, we have a quadratic equation. To solve it, we can factor it or use the quadratic formula. In this case, let's use the quadratic formula.

`\boxed{\begin{array}{l}\underline{\sf Quadratic\;Formula}\\\\x=(-b \pm √(b^2-4ac))/(2a)\\\\\textsf{when} \;ax^2+bx+c=0 \\\end{array}}`

In this case:

• a = 1
• b = -10
• c = 16

Substitute the values of a, b and c into the formula and simplify:

`x=(-(-10) \pm √((-10)^2-4(1)(16)))/(2(1))`

`x=(10 \pm √(100-64))/(2)`

`x=(10 \pm √(36))/(2)`

`x=(10 \pm √(6^2))/(2)`

`x=(10 \pm 6)/(2)`

Consider both the positive and negative square root:

`x=(10 +6)/(2)\implies x=(16)/(2)=8`

`x=(10-6)/(2)\implies x=(4)/(2)=2`

So, the solutions to the given equation are x = 8 and x = 2.

Answer 2

x = 8

x = 2

Explanation:

To solve:

`\sf √(6x -12) = x - 2`

Solution:

We can solve this in the following ways:

`\sf √(6x -12) = x - 2`

Square both sides of the equation.

`\sf (√(6x - 12))^2 = (x - 2)^2`

Expand:

`\sf 6x - 12 = x^2 - 4x + 4`

Subtract 6x and add 12 on both sides,

`\sf 6x - 12 - 6x + 12 = x^2 - 4x + 4 - 6x + 12`

`\sf 0 = x^2 - 10x + 16`

Doing middle term factorization.

`\sf x^2 - (8+2)x + 16 = 0`

`\sf x^2 - 8x - 2x + 16`

Take common from each two terms.

`\sf x(x-8)-2(x -8)`

Take common and keep remaining in the bracket.

`\sf (x -8) (x-2) = 0`

Either

x - 8 = 0

x = 8

Or

x - 2 = 0

x = 2

Therefore, the two solutions to the equation x = 8 and x = 2.

However, we need to check our solutions to make sure that they are valid. We can do this by substituting them back into the original equation.

If we substitute x = 8 into the original equation, we get:

`\sf √(6(8) - 12) = 8 - 2`

`\sf √(48 - 12) = 6`

`\sf √(36) = 6`

`\sf 6 = 6`

This is true, so x = 8 is a valid solution.

If we substitute x = 2 into the original equation, we get:

`\sf √(6(2) - 12) = 2 - 2`

`\sf √(12 - 12) = 0`

`\sf √(0) = 0`

`\sf 0 = 0`

This is also true, so x = 2 is also a valid solution.

Therefore, the two solutions to the equation are x = 8 and x = 2.