Trinomials can be factored three different ways: By grouping, With quadratic formula, Completing the square.
Quadratic formula hard to memorize but always works. Its template is: [tex]x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} [/tex]
To make quadratic formula work, we need to put the equation into the standard form. In this case, it's already in that form. [tex]a=4, b=23, c=-6[/tex].
When we plug them in: [tex]x = \frac{- 23 \pm \sqrt{ {23}^{2} - 4 \times 4 \times ( - 6) } }{2 \times 4}[/tex]
Simplify:
[tex]x = \frac{ - 23 \pm\sqrt{529 + 96} }{8} [/tex]
[tex]x = \frac{ - 23\pm \sqrt{625} }{8} [/tex]
[tex]x = \frac{ - 23 \pm25 }{8} [/tex]
Branch out the plus-minus sign:
[tex]x = \frac{ - 23 + 25}{8} \: \: \: \: \:x = \frac{ - 23 - 25}{8} [/tex]
[tex]x = \frac{2}{8} \: \: \: \: \:x = \frac{ - 48}{8} [/tex]
[tex]x = \frac{1}{4} \: \: \: \: \:x = - 6[/tex]
So it's roots are [tex] \frac{1}{4} , -6[/tex].
Roots of a polynomial are the values which made the equation equal to 0. To make this, we need to write the factored form as [tex](x - \frac{1}{4} )(x + 6)[/tex]
And because [tex] - \frac{1}{4} [/tex] is a fraction, we need to get rid of it. We can do this by multiplying the left side with 4 and our final answer would be [tex](4x - 1)(x + 6)[/tex].
