[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]
[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
Let's take the number as 'x'.
We are given that, 8x + x² = 100
Now, let's solve for x.
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[tex]8 x + x ^ { 2 } = 100[/tex]
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.
[tex]x^{2}+8x=100 [/tex]
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.
[tex]x^{2}+8x+4^{2}=100+4^{2} [/tex]
Square 4.
[tex]x^{2}+8x+16=100+16 [/tex]
Add 100 to 16.
[tex]x^{2}+8x+16=116 [/tex]
Factor x²+8x+16. In general, when x²+bx+c is a perfect square, it can always be factored as [tex]\left(x+\frac{b}{2}\right)^{2}[/tex].
[tex]\left(x+4\right)^{2}=116 [/tex]
Take the square root of both sides of the equation.
[tex]\sqrt{\left(x+4\right)^{2}}=\sqrt{116} [/tex]
Simplify.
[tex]x+4=2\sqrt{29} \\ x+4=-2\sqrt{29} [/tex]
Subtract 4 from both sides of the equation.
[tex] \huge \boxed{ \boxed{ \bf \: x=2\sqrt{29}-4 }}\\ \huge \boxed{\boxed{ \bf \: x=-2\sqrt{29}-4 }}[/tex]
