Question

How do you solve \[4{{x}^{2}}+5x-6=0\] using the quadratic formula?

Answer 1

Hint: For answering this question we are asked to solve the given equation \[4{{x}^{2}}+5x-6=0\] and find the value of the variable $x$ . For solving that we will make some transformations like shifting from right hand to left hand side and vice versa and basic arithmetic simplifications like addition and subtraction then we need to conclude the value of the unknown variable $x$ .

Complete step-by-step answer:
Now considering the question in which we are asked to solve the given equation \[4{{x}^{2}}+5x-6=0\] and derive the value of the unknown variable \[x\].
Firstly, we will observe the given equation carefully and compare the given quadratic equation with the basic quadratic equation \[a{{x}^{2}}+bx+c=0\].
By comparing the given quadratic equation \[4{{x}^{2}}+5x-6=0\] with the basic quadratic equation \[a{{x}^{2}}+bx+c=0\] , we get,
\[\Rightarrow a=4,b=5,c=-6\]
Here in the question the variable x is nothing but the roots of the given equation \[4{{x}^{2}}+5x-6=0\] .
We can find the roots of the quadratic equation by basic formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] . After comparing the given quadratic equation with the basic equation we substitute them and we get,
\[\Rightarrow x=\dfrac{-5\pm \sqrt{25-4\left( 4\times \left( -6 \right) \right)}}{2\times 4}\]
Now we will perform basic arithmetic multiplication and the equation will be reduced to \[\Rightarrow \]\[x=\dfrac{-5\pm \sqrt{25-\left( -96 \right)}}{2\times 4}\]
Here we will transform the equation by using basic arithmetic simplification of addition and equation will be transformed into
\[\Rightarrow x=\dfrac{-5\pm \sqrt{121}}{8}\]
Now we perform the basic application of square root and reduce the given equation into
\[\Rightarrow x=\dfrac{-5\pm 11}{8}\]
And now we solve the equation by using simple arithmetic division and addition and we get
\[\Rightarrow x=-2,\dfrac{3}{4}\]
Therefore we can conclude that the value of \[x\] in \[4{{x}^{2}}+5x-6=0\] is \[x=-2,\dfrac{3}{4}\].

Note: We should be very peculiar with our calculations while solving the quadratic equations with simple basic concepts of arithmetic simplifications and transformations. Similarly questions of these kind quadratic equations can be solved by using the basic formula of \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] which are the roots of the given quadratic equation. This question can also be solved using factorization but here we have been asked to use the quadratic formula. The solution using factorisation will be as follows $4{{x}^{2}}+5x-6=0\Rightarrow 4{{x}^{2}}-3x+8x-6=0\Rightarrow \left( x+2 \right)\left( 4x-3 \right)=0\Rightarrow x=-2,\dfrac{3}{4}$ .