Question

How do you solve $(4a - 7)(3a + 8) = 0$?

Answer 1

Hint: Here we will find the product of two brackets in which there are two terms which will give us four terms then finding the like terms in it and simplify the expression and last will find the value for the unknown term “a”.

Complete step by step solution:
Take the given expression: $(4a - 7)(3a + 8) = 0$
Find the product of two brackets.
$ \Rightarrow 4a(3a + 8) - 7(3a + 8) = 0$
Multiply the term outside the bracket with all the terms inside the bracket, when there is a positive term outside the bracket then there will be no change in the sign of the terms inside the bracket whereas if there is negative sign outside the bracket then the sign of all the terms inside the bracket changes.
$ \Rightarrow 12{a^2} + 32a - 21a - 56 = 0$
Simplify among the like terms.
$ \Rightarrow 12{a^2} + \underline {32a - 21a} - 56 = 0$
$ \Rightarrow 12{a^2} + 11a - 56 = 0$
Compare the above equation with the standard equation: $a{x^2} + bx + c = 0$
$
   \Rightarrow a = 12 \\
   \Rightarrow b = 11 \\
   \Rightarrow c = - 56 \;
 $
Also, $\Delta = {b^2} - 4ac$
Place the values from the given comparison
$ \Rightarrow \Delta = {(11)^2} - 4(12)( - 56)$
Simplify the above equation –
$ \Rightarrow \Delta = 121 + 2688$
Do subtraction –
$ \Rightarrow \Delta = 2809$
Take square root on both the sides of the equation –
$ \Rightarrow \sqrt \Delta = \sqrt {2809} $
The above equation can be re-written as –
$ \Rightarrow \sqrt \Delta = \sqrt {{{53}^2}} $
Simplify the above equation applying the square of the known number.
$ \Rightarrow \sqrt \Delta = 53$
Now, roots of the given equation can be expressed as –
 \[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\]
Place values in the above equation –
 \[x = \dfrac{{ - (11) \pm 53}}{{2(12)}}\]
Product of minus and minus is plus. Simplify the above equation –
 \[x = \dfrac{{ - 11 + 53}}{{24}}\] or \[x = \dfrac{{ - 11 - 53}}{{24}}\]
Simplify the above equations.
 \[x = \dfrac{{42}}{{24}}\] or \[x = - \dfrac{{64}}{{24}}\]
Common factors from the numerator and the denominator cancel each other.
 \[x = \dfrac{7}{4}\] or \[x = - \dfrac{8}{3}\]
Therefore \[x = \dfrac{7}{4}\] or \[x = - \dfrac{8}{3}\]
This is the required solution.
So, the correct answer is “ \[x = \dfrac{7}{4}\] or \[x = - \dfrac{8}{3}\] ”.

Note: Be careful regarding the sign convention. Always remember that the square of negative number or the positive number is always positive. Also, product of two negative numbers is always positive whereas, product of one positive and one negative number gives us the negative number.