Question

Solve the following equation and calculate the answer to two decimal places.
${{x}^{2}}-5x-10=0$.

Answer 1

Hint: This is a quadratic equation so we will solve this with the help of quadratic formula. The quadratic formula for the equation $a{{x}^{2}}+bx+c=0$ is
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

Complete step-by-step answer:
In the question, we are given a quadratic polynomial on the left-hand side. Quadratic polynomial is a type of polynomial which has the highest degree of x as 2. The general form of the quadratic equation is $a{{x}^{2}}+bx+c=0$. The question demands that we have to found those values of x at which the quadratic equation becomes zero. These values of x are called zeros or roots of the above equation, we are going to use the above quadratic formula. The quadratic formula is given as: -
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Where x are the zeroes of the quadratic equation, a is the coefficient of ${{x}^{2}}$, b is the coefficient of x and c is the constant term. In our case, we have $a=1$, $b=-5$ and $c=-10$. Now, we will put these values in the quadratic formula to obtain the zeroes.
$x=\dfrac{5\pm \sqrt{{{\left( 5 \right)}^{2}}-4\left( 1 \right)\left( -10 \right)}}{2\left( 1 \right)}$
$\Rightarrow x=\dfrac{5\pm \sqrt{65}}{2}$
The value of $\sqrt{65}$ up to 3 decimal places is 8.062. We will put this value of $\sqrt{65}$ in the above quadratic formula. After putting the value, we will get;
$x=\dfrac{5.000\pm 8.062}{2}$
$\Rightarrow x=\dfrac{5.000}{2}\pm \dfrac{8.062}{2}$
$\Rightarrow x=2.500\pm 4.031$
Here, let us consider that there is a plus sign. Thus, the value of x will be: -
$\Rightarrow x=2.500+4.031$
$\Rightarrow x=6.531$
But we have to give value up to two decimal places, thus,
$\Rightarrow x=6.53$
Now, in another case, let us consider that there is a minus sign. Thus, the value of x becomes: -
$\Rightarrow x=2.500-4.031$
$\Rightarrow x=-1.531$
$\Rightarrow x=-1.53$
Thus, the two roots of the equation are: -
$x=6.53$ and $x=-1.53$

Note: We can only use the quadratic formula method to solve the quadratic equation given in question. We cannot use the factorization method because we will not be able to do factors in this case. Similarly, we can’t use a perfect square method because we will not be able to form a perfect square precisely.