When 6 is subtracted from the square of a number, the result is 5 times the number, then the negative solution is -1
<h3><u>Solution:</u></h3>
Given that when 6 is subtracted from the square of a number, the result is 5 times the number
To find: negative solution
Let "a" be the unknown number
Let us analyse the given sentence
square of a number = ![a^2](https://tex.z-dn.net/?f=a%5E2)
6 is subtracted from the square of a number = ![a^2 - 6](https://tex.z-dn.net/?f=a%5E2%20-%206)
5 times the number = ![5 \times a](https://tex.z-dn.net/?f=5%20%5Ctimes%20a)
<em><u>So we can frame a equation as:</u></em>
6 is subtracted from the square of a number = 5 times the number
![a^2 - 6 = 5 \times a\\\\a^2 -6 -5a = 0\\\\a^2 -5a -6 = 0](https://tex.z-dn.net/?f=a%5E2%20-%206%20%3D%205%20%5Ctimes%20a%5C%5C%5C%5Ca%5E2%20-6%20-5a%20%3D%200%5C%5C%5C%5Ca%5E2%20-5a%20-6%20%3D%200)
<em><u>Let us solve the above quadratic equation</u></em>
For a quadratic equation
where ![a \neq 0](https://tex.z-dn.net/?f=a%20%5Cneq%200)
![x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-b%20%5Cpm%20%5Csqrt%7Bb%5E%7B2%7D-4%20a%20c%7D%7D%7B2%20a%7D)
Here in this problem,
![a^2-5 a-6=0 \text { we have } a=1 \text { and } b=-5 \text { and } c=-6](https://tex.z-dn.net/?f=a%5E2-5%20a-6%3D0%20%5Ctext%20%7B%20we%20have%20%7D%20a%3D1%20%5Ctext%20%7B%20and%20%7D%20b%3D-5%20%5Ctext%20%7B%20and%20%7D%20c%3D-6)
Substituting the values in above quadratic formula, we get
![\begin{array}{l}{a=\frac{-(-5) \pm \sqrt{(-5)^{2}-4(1)(-6)}}{2 \times 1}} \\\\ {a=\frac{5 \pm \sqrt{25+16}}{2}=\frac{5 \pm \sqrt{49}}{2}} \\\\ {a=\frac{5 \pm 7}{2}}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7Ba%3D%5Cfrac%7B-%28-5%29%20%5Cpm%20%5Csqrt%7B%28-5%29%5E%7B2%7D-4%281%29%28-6%29%7D%7D%7B2%20%5Ctimes%201%7D%7D%20%5C%5C%5C%5C%20%7Ba%3D%5Cfrac%7B5%20%5Cpm%20%5Csqrt%7B25%2B16%7D%7D%7B2%7D%3D%5Cfrac%7B5%20%5Cpm%20%5Csqrt%7B49%7D%7D%7B2%7D%7D%20%5C%5C%5C%5C%20%7Ba%3D%5Cfrac%7B5%20%5Cpm%207%7D%7B2%7D%7D%5Cend%7Barray%7D)
We have two solutions for "a"
![\begin{array}{l}{a=\frac{5+7}{2} \text { and } a=\frac{5-7}{2}} \\\\ {a=\frac{12}{2} \text { and } a=\frac{-2}{2}}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7Ba%3D%5Cfrac%7B5%2B7%7D%7B2%7D%20%5Ctext%20%7B%20and%20%7D%20a%3D%5Cfrac%7B5-7%7D%7B2%7D%7D%20%5C%5C%5C%5C%20%7Ba%3D%5Cfrac%7B12%7D%7B2%7D%20%5Ctext%20%7B%20and%20%7D%20a%3D%5Cfrac%7B-2%7D%7B2%7D%7D%5Cend%7Barray%7D)
<h3>a = 6 or a = -1</h3>
We have asked negative solution. So a = -1
Thus the negative solution is -1