Question

How do you solve \[7{x^2} - 63 = 0\]?

Answer 1

Hint: Write the given quadratic equation in the standard form \[a{x^2} + bx + c = 0\]. Use the quadratic formula to solve and then obtain the roots. The quadratic formula for the standard quadratic equation is \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].

Complete step by step solution:
We are given that we are required to solve \[7{x^2} - 63 = 0\].
We can write this as: -
\[ \Rightarrow 7{x^2} + 0x - 63 = 0\]
We know that the general quadratic equation is given by \[a{x^2} + bx + c = 0\], where a, b and c are constants.
Now, we know that its roots are given by the following expression: -
\[ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Comparing the general equation \[a{x^2} + bx + c = 0\] with the given equation \[7{x^2} + 0x - 63 = 0\], we will then obtain the following:-
\[ \Rightarrow \]a = 7, b = 0 and c = - 63
Now, putting these in the formula given by \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we will then obtain the following expression: -
\[ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {{{(0)}^2} - 4(7)( - 63)} }}{{2(7)}}\]
Simplifying the calculations in the right hand side of the above mentioned expression, we will then obtain the following expression: -
\[ \Rightarrow x = \pm \dfrac{{\sqrt {4 \times 7 \times 63} }}{{2 \times 7}}\]
\[ \Rightarrow x = \pm \dfrac{{\sqrt {2 \times 2 \times 7 \times 7 \times 3 \times 3} }}{{2 \times 7}}\]
Simplifying the calculations in the right hand side of the above mentioned expression further, we will then obtain the following expression: -
\[ \Rightarrow x = \pm \dfrac{{2 \times 7 \times 3}}{{2 \times 7}}\]
Simplifying the right hand side of the above expression, we will then obtain the following expression: -
\[ \Rightarrow x = - 3,3\]
Thus, the required solutions or roots for the given quadratic equation are \[3\] and \[ - 3\].

Note:
The students must note that there is an alternate way to solve the same question done as follows. Here, we will use basic algebraic rules.
We can write the given equation as follows: -
\[ \Rightarrow 7{x^2} - 63 = 0\]
Taking 7 common from first two terms, we will then obtain the following equation with us: -
\[ \Rightarrow 7\left( {{x^2} - 9} \right) = 0\]
Divide both sides of the equation by the number 7 and simplify as follows: -
\[ \Rightarrow \dfrac{{7\left( {{x^2} - 9} \right)}}{7} = \dfrac{0}{7}\]
\[ \Rightarrow \left( {{x^2} - 9} \right) = 0\]
Now, express the second term 9 as \[{3^2}\] and use the algebraic identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] as follows: -
\[ \Rightarrow \left( {x - 3} \right)\left( {x + 3} \right) = 0\]
Thus, it implies that variable x has two solutions one is positive 3 and other is negative 3.