Question

Solve the given quadratic equation ${x^2} - x - 650 = 0$

Answer 1

Hint: The given problem requires us to solve a quadratic equation. There are various methods that can be employed to solve a quadratic equation like completing the square method, using quadratic formula and by splitting the middle term. Using the quadratic formula gives us the roots of the equation directly with ease. We will first compare the given equation with the general form of the quadratic equation to find the coefficients of the terms that are to be used while solving the equation using quadratic formula.

Complete step-by-step solution:
In the given question, we are required to solve the equation ${x^2} - x - 650 = 0$ with the help of a quadratic formula.
The quadratic formula can be employed for solving an equation only if we compare the given equation with the standard form of quadratic equation.
Comparing with standard quadratic equation $a{x^2} + bx + c = 0$
Here,$a = 1$, $b = - 1$ and $c = - 650$.
Now, using the quadratic formula, we get the roots of the equation as:
$x = \dfrac{{( - b) \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Substituting the values of a, b, and c in the quadratic formula, we get,
$x = \dfrac{{ - \left( { - 1} \right) \pm \sqrt {{{\left( { - 1} \right)}^2} - 4 \times 1 \times \left( { - 650} \right)} }}{{2 \times 1}}$
Opening the brackets,
$ \Rightarrow x = \dfrac{{1 \pm \sqrt {1 + 4 \times 650} }}{{2 \times 1}}$
Simplifying the calculations, we get,
$ \Rightarrow x = \dfrac{{1 \pm \sqrt {1 + 2600} }}{{2 \times 1}}$
Adding the like terms,
$ \Rightarrow x = \dfrac{{1 \pm \sqrt {2601} }}{{2 \times 1}}$
Computing the square root of $2601$,
$ \Rightarrow x = \dfrac{{1 \pm 51}}{2}$
Now, we take the positive and negative signs one by one to get both the roots of the equation.
So, taking the positive sign first, we get,
$x = \dfrac{{1 + 51}}{2}$
$ \Rightarrow x = \dfrac{{52}}{2}$
Cancelling common factors in numerator and denominator, we get,
$ \Rightarrow x = 26$
So, one root of the quadratic equation ${x^2} - x - 650 = 0$ is $26$.
Now, considering the negative sign, we get,
$x = \dfrac{{1 - 51}}{2}$
Simplifying the calculations, we get,
$ \Rightarrow x = \dfrac{{ - 50}}{2}$
Cancelling common factors in numerator and denominator, we get,
$ \Rightarrow x = - 25$
So, we get the second root as $ - 25$.
So, the roots of the equation ${x^2} - x - 650 = 0$ are: $x = - 25$ and $x = 26$.

Note: Quadratic formula is the easiest and most efficient formula to calculate the roots of an equation. We must know the standard form of the quadratic equations as $a{x^2} + bx + c = 0$ so as to compare the given equation and find the values of coefficients of terms. We must take care while cancelling common factors in numerator and denominator and always report the final answer in simplest terms.