Question

What are the roots of the equatio
x² - x -42=0

Answer 1

The roots are 7 and -6.

Keys:

• 
  `ax^2+bx+c=0`
• 
  `x_(1,\:2)=(-b\pm √(b^2-4ac))/(2a)`
• 
  `-(-a)^(^n^) = a^(^n^)`
• 
  `\sqrt[n]{a}^n = a`
• 
  `1^a = 1`

When you see a ± in a quadratic equation, you will have multiple solutions; more than one.

Explanation:

solve the main expression

`x^2-x-42=0\\x_(1,\:2)=(-\left(-1\right)\pm √(\left(-1\right)^2-4\cdot \:1\cdot \left(-42\right)))/(2\cdot \:1)\\\rightarrow √(\left(-1\right)^2-4\cdot \:1\cdot \left(-42\right))\\=√(\left(-1\right)^2+4\cdot \:1\cdot \:42)\\\rightarrow \left(-1\right)^2\\=1^2\\=1\\\rightarrow 4\cdot \:1\cdot \:42\\=168\\=√(1+168)\\=√(169)\\=√(13^2)\\=13\\x_(1,\:2)=(-\left(-1\right)\pm \:13)/(2\cdot \:1)`

solve for x₁

`(-\left(-1\right)+13)/(2\cdot \:1)\\=(1+13)/(2\cdot \:1)\\=(14)/(2\cdot \:1)\\=(14)/(2)\\=7`

solve for x₂

`(-\left(-1\right)-13)/(2\cdot \:1)\\=(1-13)/(2\cdot \:1)\\=(-12)/(2\cdot \:1)\\=(-12)/(2)\\=-(12)/(2)\\=-6`

Answer 2

`x^2-x-42=0\\x^2+6x-7x-42=0\\x(x+6)-7(x+6)=0\\(x-7)(x+6)=0\\x=7 \vee x=-6`