Question

How do you solve \[2{{x}^{2}}-5x-10=0\] using the quadratic formula?

Answer 1

Hint: The given equation will be solved using the quadratic formula as asked in the question. The given equation is in the standard form, \[a{{x}^{2}}+bx+c=0\].To check whether the roots are real or imaginary we will find the discriminant of the given equation which is \[D={{b}^{2}}-4ac\] and we get it as \[105>0\]. So, on substituting the values of a, b, c in the quadratic formula, we get two real roots for the equation \[2{{x}^{2}}-5x-10=0\].

Complete step by step solution:
Quadratic formula is of the form \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] where a, b and c are from the quadratic equation \[a{{x}^{2}}+bx+c=0\].
According to the given question, we have to solve the given equation for \[x\] using the quadratic formula
We will start with finding the discriminant of the given equation,
\[2{{x}^{2}}-5x-10=0\]-----(1)
We know that
\[D={{b}^{2}}-4ac\]
On comparing the discriminant with the equation (1), we get the value of variables as,
\[a=2,b=-5,c=-10\]
On substituting these values in the discriminant formula, we get,
\[\Rightarrow D={{(-5)}^{2}}-4(2)(-10)\]
\[\Rightarrow D=25+80\]
\[\Rightarrow D=105>0\]
We have the value of \[D>0\], therefore the given equation has 2 real roots. We will use the quadratic formula to find the values of \[x\].
We have the quadratic formula as,
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Substituting the values of variables, we have, we get,
\[\Rightarrow x=\dfrac{-(-5)\pm \sqrt{(105)}}{2(2)}\]
\[\Rightarrow x=\dfrac{5\pm \sqrt{(105)}}{4}\]
\[\Rightarrow x=\dfrac{5+\sqrt{(105)}}{4},\dfrac{5-\sqrt{(105)}}{4}\]

Therefore, we got the two values of \[x=\dfrac{5+\sqrt{(105)}}{4},\dfrac{5-\sqrt{(105)}}{4}\].

Note: Quadratic formula that we used in our solution is a versatile equation and can be used for finding both real roots as well as imaginary roots as well. So, no matter what type of quadratic equation we get, the quadratic formula always works. The only thing we need to keep in mind while using the quadratic formula is the complexity that we have to deal with and is advisable to carry out each calculation step wise. The discriminant that we use, to check the number of values as well as the type of value is also extracted from this quadratic formula.
That is, \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] and \[D={{b}^{2}}-4ac\]
This can also be written as, \[x=\dfrac{-b\pm \sqrt{D}}{2a}\]