Question

If the square of a number is added to 8 times the number, the result is 100.​

Answer 1

`\huge \boxed{\mathbb{QUESTION} \downarrow}`

• If the square of a number is added to 8 times the number, the result is 100. Find x.

`\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}`

Let's take the number as 'x'.

• Square of x = x²
• 8 times x = 8x

We are given that, 8x + x² = 100

Now, let's solve for x.

__________________

`8 x + x ^ { 2 } = 100`

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x²+bx=c.

`x^(2)+8x=100`

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left-hand side of the equation a perfect square.

`x^(2)+8x+4^(2)=100+4^(2)`

Square 4.

`x^(2)+8x+16=100+16`

Add 100 to 16.

`x^(2)+8x+16=116`

Factor x²+8x+16. In general, when x²+bx+c is a perfect square, it can always be factored as
`\left(x+(b)/(2)\right)^(2)`.

`\left(x+4\right)^(2)=116`

Take the square root of both sides of the equation.

`\sqrt{\left(x+4\right)^(2)}=√(116)`

Simplify.

`x+4=2√(29) \\ x+4=-2√(29)`

Subtract 4 from both sides of the equation.

`\huge \boxed{ \boxed{ \bf \: x=2√(29)-4 }}\\ \huge \boxed{\boxed{ \bf \: x=-2√(29)-4 }}`

• x can be either of these values.