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 $\left(x+\frac{b}{2}\right)^{2}$.

$\left(x+4\right)^{2}=116$

Take the square root of both sides of the equation.

$\sqrt{\left(x+4\right)^{2}}=\sqrt{116}$

Simplify.

$x+4=2\sqrt{29} \\ x+4=-2\sqrt{29}$

Subtract 4 from both sides of the equation.

$\huge \boxed{ \boxed{ \bf \: x=2\sqrt{29}-4 }}\\ \huge \boxed{\boxed{ \bf \: x=-2\sqrt{29}-4 }}$

• x can be either of these values.