x = -3/2 or x = 5/3
Explanation:Given:
[tex]6)\text{ 6x}^2\text{ = x + 15}[/tex]To find:
the value of x
First we need to re-write the given equation into the form ax² + bx + c = 0:
a = coefficient of x², b = coefficient of b and c = constant
[tex]\begin{gathered} subtract\text{ x from both sides} \\ 6x^2\text{ -x = x - x + 15 } \\ 6x^2\text{ - x = 15} \\ subtract\text{ 15 from both sides} \\ 6x^2\text{ - x - 15 = 15 - 15} \\ 6x^2\text{ -x - 15 = 0} \end{gathered}[/tex]a = 6, b = -1, c = -15
Next we will find the factors of ac whose sum gives b
ac = a × c = 6 × -15 = -90
So we need to find factors of -90 whose sum will give -1
The factors are: -10 and 9
-10 × 9 = -90
-10 + 9 = -1
[tex]\begin{gathered} 6x^2\text{ - x - 15 = 0 will become} \\ 6x^2\text{ - 10x + 9x - 15 = 0} \end{gathered}[/tex][tex]\begin{gathered} we\text{ will factorise by bringing out the common numbers and variables:} \\ 2x\text{ is common to 6x}^2\text{ and 10x} \\ 6x^2=\text{ 2x\lparen3x\rparen} \\ 10x\text{ = 2x\lparen5\rparen} \\ factoring\text{ }6x^2\text{ - 10xwill give 2x\lparen3x + 5\rparen} \\ \\ 3\text{ is common to 9x and -15} \\ 9x\text{ = 3\lparen3x\rparen} \\ -15\text{ = 3\lparen-5\rparen} \\ \\ 6x^2\text{ -10x + 9x - 15 = 0} \\ 2x(3x\text{ - 5\rparen + 3\lparen3x - 5\rparen = 0} \end{gathered}[/tex]NB: when we factorise using factoring method, the numbers in both parenthesis will be the same
[tex]\begin{gathered} (2x\text{ + 3\rparen\lparen3x - 5\rparen = 0} \\ 2x\text{ +3 = 0 or 3x - 5 = 0} \\ 2x\text{ = -3 or 3x = 5} \\ x\text{ = -3/2 or x = 5/3} \end{gathered}[/tex]