Question

Solve the following quadratic equation:
\[5{{x}^{2}}-4\sqrt{2}x-1=0\]

Answer 1

Hint: In algebra, a quadratic equation is any equation that can be rearranged in standard form as \[a{{x}^{2}}+bx+c=0\], where x represents an unknown variable, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no \[a{{x}^{2}}\] term. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.
In such questions, the word ‘solve’ means to get the roots of the given equation.
The quadratic formula that would be used in the question to get the roots of a quadratic equation is as follows
\[roots=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]

Complete step-by-step answer:
As mentioned in the question, we have to find the roots of the quadratic equation that is given.
The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is no real solution, there are two complex solutions. If there is only one solution, one says that it is a double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation \[a{{x}^{2}}+bx+c=a(x-r)(x-s)=0\] where r and s are the solutions for x.

Now, we will use the quadratic formula to solve the question which is given in the hint as follows
\[\begin{align}
  & \Rightarrow 5{{x}^{2}}-4\sqrt{2}x-1=0 \\
 & \Rightarrow roots=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
 & roots=\dfrac{4\sqrt{2}\pm \sqrt{{{\left( -4\sqrt{2} \right)}^{2}}+4\times 5\times 1}}{2\times 5} \\
 & \Rightarrow roots=\dfrac{4\sqrt{2}\pm \sqrt{32+20}}{10} \\
 & \Rightarrow roots=\dfrac{4\sqrt{2}\pm \sqrt{52}}{10} \\
 & \Rightarrow roots=\dfrac{2\sqrt{2}\pm \sqrt{13}}{5} \\
 & \Rightarrow roots=\dfrac{2\sqrt{2}+\sqrt{13}}{5},\dfrac{2\sqrt{2}-\sqrt{13}}{5} \\
\end{align}\]
Hence, these are the roots of the given quadratic equation.

NOTE: For finding the roots of a given equation, we can follow three methods which are as follows:-
1. Using quadratic formula
2. Completing the square
3. Splitting the middle term
But it is advisable to use the quadratic formula as it is more convenient.
Also, the calculations in this question’s solution part should be done very carefully as there are chances that the students might commit a mistake.