Question

If the square of a number is added to 8 times the nuber the result is 100

Answer 1

Solution :-

• Let's take the number as 'x'.

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• Square of x = x²
• 8 times x = 8x

• We are given that, 8x + x² = 100

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$\purple{\bf:\implies {8 x + x ^ { 2 } = 100}}$

• 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.

$\tt :\implies {x^{2}+8x=100}$

• 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.

$\tt :\implies {x^{2}+8x+4^{2}=100+4^{2} }$

$\tt :\implies {x^{2}+8x+16=100+16 }$

$\tt :\implies {x^{2}+8x+16=116}$

• Factor x²+8x+16. In general, when x²+bx+c is a perfect square, it can always be factored as $\tt{\left(x+\dfrac{b}{2}\right)^{2}}$.

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

• Take the square root of both sides of the equation.

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

Simplify.

$\tt :\implies {x+4=2\sqrt{29} }$

• Subtract 4 from both sides of the equation.

$\purple{\bf:\implies {x =2\sqrt{29} -4 }}$