Question

How do you solve \[6{x^2} - 21x + 15 = 0\]?

Answer 1

Hint: Here in this question, we have to solve the given equation, the given equation is in the form of a quadratic equation. This is a quadratic equation for the variable x. By using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we can determine the solutions.

Complete step-by-step solution:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factoring or by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. consider the given equation \[6{x^2} - 21x + 15 = 0\].
In general, the quadratic equation is represented as \[a{x^2} + bx + c = 0\], when we compare the above equation to the general form of equation the values are as follows. a=6 b=-21 and c=15. Now substituting these values to the formula for obtaining the roots we have
\[x = \dfrac{{ - ( - 21) \pm \sqrt {{{( - 21)}^2} - 4(6)(15)} }}{{2(6)}}\]
On simplifying the terms, we have
\[ \Rightarrow x = \dfrac{{21 \pm \sqrt {441 - 360} }}{{12}}\]
Now subtract 360 from 441 we get
\[ \Rightarrow x = \dfrac{{ - 21 \pm \sqrt {81} }}{{12}}\]
The number 81 is a perfect square number and we have a square root for this. So, the square root of 81 is 9 and so we have.
Therefore, we have \[x = \dfrac{{ - 21 + 9}}{{12}}\] or \[x = \dfrac{{ - 21 - 9}}{{12}}\]. We can simplify for further so we get
\[ \Rightarrow x = \dfrac{{ - 12}}{{12}}\] and \[x = \dfrac{{ - 30}}{{12}}\]
So we have
\[ \Rightarrow x = - 1\] and \[x = - 2.5\]
hence we have solved the quadratic equation and found the value of the variable x.
The equation is also solved by using the factorisation method.

Note: The quadratic equation can be solved by using the factorisation method and we also find the roots by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.