Question

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

Answer 1

Hint: We will be using the quadratic formula to solve the above expression. So, first we will rewrite the equation in the standard form, that is, \[a{{x}^{2}}+bx+c=0\] and so we have, \[2{{x}^{2}}+5x+4=0\]. Now, we will check if the equation has real roots or not. For that we will use the discriminant. Applying the discriminant, we get \[D<0\], that means the equation has no real roots or we can say that the equation has 2 imaginary roots. We will use iota (\[i\]), which has the value \[i=\sqrt{(-1)}\], in the quadratic formula to get the value of \[x\], we get the value as \[\dfrac{-5+\sqrt{7}i}{4},\dfrac{-5-\sqrt{7}i}{4}\]

Complete step by step solution:
According to the given question, we have been asked to solve the above equation for \[x\] using the quadratic formula.
The given expression is, \[2{{x}^{2}}+5x=-4\], so we will start with re-writing the given equation into the standard form of a quadratic equation which is \[a{{x}^{2}}+bx+c=0\], we get,
\[2{{x}^{2}}+5x+4=0\]------(1)
Before applying the quadratic formula on the above expression, we will check what type of roots do the expression using discriminant, denoted by \[D\],
We know that
\[D={{b}^{2}}-4ac\]
On comparing the discriminant with the equation (1), we get the value of variables as,
\[a=2,b=5,c=4\]
On substituting these values in the discriminant formula, we get,
\[\Rightarrow D={{(5)}^{2}}-4(2)(4)\]
\[\Rightarrow D=25-32\]
\[\Rightarrow D=-7<0\]
Since, we have the value of \[D<0\], therefore the given equation has no real roots or we can also say that the given equation has 2 imaginary roots.
We will use quadratic formula to proceed further, we have,
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Substituting the values of the variables, we get,
\[\Rightarrow x=\dfrac{-(5)\pm \sqrt{(-7)}}{2(2)}\]
If we look at the root, there is a negative sign which we cannot simply solve. So, we will use a term called iota (\[i\]), which has the value \[i=\sqrt{(-1)}\], when we substitute it in the above expression, we get,
\[\Rightarrow x=\dfrac{-5\pm \sqrt{(7})i}{4}\]
\[\Rightarrow x=\dfrac{-5\pm \sqrt{7}i}{4}\]
\[\Rightarrow x=\dfrac{-5+\sqrt{7}i}{4},\dfrac{-5-\sqrt{7}i}{4}\]

Therefore, the two imaginary (or complex) values of \[x=\dfrac{-5+\sqrt{7}i}{4},\dfrac{-5-\sqrt{7}i}{4}\].

Note: The discriminant used in the above solution can have three possibilities. They are as follows:
\[D={{b}^{2}}-4ac\]
1) When \[D>0\], then the equation has 2 real roots.
2) When \[D=0\], then the equation has 1 real root.
3) When \[D<0\], then the equation has 2 imaginary roots.