Question

Solve the following quadratic equation \[{{x}^{2}}-\dfrac{3x}{10}-\dfrac{1}{10}=0\]

Answer 1

Hint: In this question firstly we have given that the equation is quadratic equation as it has the degree of two and then we will compare the given equation from general quadratic equation, then we will find out the values of \[a,b\] and \[c\] after that we will apply the quadratic formula so that we can obtain the result.

Complete step-by-step solution:
The term "quadratic" comes from the word "quad," which means "square." In other words, a quadratic equation is a "degree two equation." A quadratic equation is employed in a variety of situations. A quadratic equation also has a wide range of applications in physics, engineering, and astronomy.
The Quadratic Formula is the most straightforward method for determining the roots of a quadratic equation. There are some quadratic equations that are difficult to factor, and in these cases, we can utilise this quadratic formula to get the roots as quickly as feasible. The sum of the roots and the product of the roots of the quadratic equation can also be found using the roots of the quadratic equation. The quadratic formula's two roots are provided as a single equation. The two unique roots of the equation can be obtained using either the positive or negative sign.
The two values of \[x\] obtained by solving the quadratic equation are the roots of a quadratic equation. The zeros of the equation are also known as the roots of the quadratic equation.
The term \[{{b}^{2}}-4ac\] in the quadratic formula is known as the discriminant of a quadratic equation. The discriminant of a quadratic equation reveals the nature of roots.
Now according to the question we have given a quadratic equation that is \[{{x}^{2}}-\dfrac{3x}{10}-\dfrac{1}{10}=0\]
Taking the LCM to make the equation simple:
\[\Rightarrow \dfrac{10{{x}^{2}}-3x-1}{10}=0\]
\[\Rightarrow 10{{x}^{2}}-3x-1=0\]
Compare this given quadratic equation by general equation \[a{{x}^{2}}+bx+c=0\] so that we can get the values of \[a,b\] and \[c\]
\[\Rightarrow a=10,b=-3,c=-1\]
Now apply the quadratic formula:
\[\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Put the values of \[a,b\] and \[c\] in the quadratic formula:
\[\Rightarrow x=\dfrac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\times 10\times \left( -1 \right)}}{2\times 10}\]
\[\Rightarrow x=\dfrac{3\pm \sqrt{9+40}}{20}\]
\[\Rightarrow x=\dfrac{3\pm \sqrt{49}}{20}\]
\[\Rightarrow x=\dfrac{3\pm 7}{20}\]
We have obtained two complex roots:
\[\Rightarrow x=\dfrac{3+7}{20},\dfrac{3-7}{20}\]
\[\Rightarrow x=\dfrac{10}{20},\dfrac{-4}{20}\]
\[\Rightarrow x=\dfrac{1}{2},\dfrac{-1}{5}\]

Note: Students should know that the quadratic function's curve is shaped like a parabola. \[{{b}^{2}}-4ac\] provides the discriminant. This is used to determine the nature of a quadratic function's solutions. The values of quadratic functions can be simply determined from the input values.