Question

How do you solve \[2{{x}^{2}}+4x+10\] using quadratic formula?

Answer 1

Hint: We will solve this problem using the quadratic equation concept. As in the question it is given that we have to solve it using quadratic formula. We have to calculate the discriminant according to the quadratic formula. Then we have to check whether real roots are possible or not. If possible we will find out according to formula. If not we have to find the imaginary roots.

Complete step-by-step answer:
Let us discuss the imaginary roots.
In quadratic equations, imaginary numbers occur when the value under the radical of the quadratic formula is negative. When this occurs, the equation has no roots in the set of real numbers. The roots belong to the set of complex numbers, and will be called imaginary roots. These imaginary roots will be expressed in the form \[a\pm bi\].
Before going to solve the equation we have to know the quadratic formula
For the standard form of quadratic equation \[a{{x}^{2}}+bx+c\] the roots are
\[\left\{ \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a},\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right\}\] and the discriminant is \[{{b}^{2}}-4ac\]
Now let's start solving the given quadratic equation.
Given equation is
\[2{{x}^{2}}+4x+10\]
 Using above formula we can say that
\[a=2\]
\[b=4\]
\[c=10\]
So we have the co-efficients to find the roots.
Now we have to calculate the discriminant
Discriminant = \[{{b}^{2}}-4ac\]
Substituting the values we have in the discriminant we gt
\[\Rightarrow {{\left( 4 \right)}^{2}}-4\times 2\times 10\]
\[\Rightarrow 16-80\]
\[\Rightarrow -64\]
So the discriminant is \[-64\]
Now we have to substitute this values in the formula we will get
\[\Rightarrow \left\{ \dfrac{-4+\sqrt{-64}}{2\times 2},\dfrac{-4-\sqrt{-64}}{2\times 2} \right\}\]
We can see the value under the square root is negative; we cannot get real roots. So we have to get imaginary roots for the equation. As already said imaginary roots will be in the form of \[a\pm bi\].
Now we have found imaginary roots.
We all know that the square root of \[64\] is \[8\]. But we have \[-64\] so its square root can be represented as \[8i\].
After substituting \[4i\] the roots will look like
\[\Rightarrow \left\{ \dfrac{-4+8i}{2\times 2},\dfrac{-4-8i}{2\times 2} \right\}\]
By simplifying it we will get
\[\Rightarrow \left\{ \dfrac{-4+8i}{4},\dfrac{-4-8i}{4} \right\}\]
From all the terms we can take \[4\] as common in numerator
\[\Rightarrow \left\{ \dfrac{4\left( -1+2i \right)}{4},\dfrac{4\left( -1-2i \right)}{4} \right\}\]
Now we can cancel both \[4's\] in numerator and denominator.
\[\Rightarrow \left\{ -1+2i,-1-2i \right\}\]
 So the roots of the given equation \[2{{x}^{2}}+4x+10\] are \[\left\{ -1+2i,-1-2i \right\}\]
It can also be represented as \[-1\pm 2i\].

Note: we can also solve this question by taking \[2\] as common in the equation in the beginning and solve for roots. It will help us to reduce the calculation part. Here we have taken \[\sqrt{-64}\] as \[8i\] because we represent negative radicals with\[{{i}^{2}}\] .
\[\sqrt{-64}=\sqrt{{{i}^{2}}\times 64}=\sqrt{i}\times \sqrt{64}=8i\]
So we can calculate the square root of any negative number like this.