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Question: If \(t > 0\) and \({t^2} - 4 = 0\), what is the value of t? (A) \(1\) (B) \(2\) (C) \( - 1\) ...

If t>0t > 0 and t2−4=0{t^2} - 4 = 0, what is the value of t?
(A) 11
(B) 22
(C) −1 - 1
(D) −2 - 2

Explanation

Solution

The given problem requires us to solve a quadratic equation. There are various methods that can be employed to solve a quadratic equation such as completing the square method, using quadratic formula and by splitting the middle term. We can also use some of the algebraic identities such as a2−b2=(a−b)(a+b){a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right) to factorize a quadratic equation.

Complete step-by-step solution:
In the given question, we are required to solve the given equation t2−4=0{t^2} - 4 = 0 given the condition t>0t > 0.
Consider the quadratic equation t2−4=0{t^2} - 4 = 0.
The equation could have been solved by various methods such as completing the square method, splitting the middle term and using the quadratic formula. But we will apply the algebraic identity a2−b2=(a−b)(a+b){a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right) to factorize the quadratic equation. As we know that we can express 44 as the square of the number 22. So, we have,
⇒(t−2)(t+2)=0\Rightarrow \left( {t - 2} \right)\left( {t + 2} \right) = 0
Now, the equation is in factored form and we just need to equate each of its factors to zero to find the required roots of the equation.
Since the product of two factors is zero. So, either of the two factors have to be equal to zero.
So, we have, either (t−2)=0\left( {t - 2} \right) = 0 or (t+2)=0\left( {t + 2} \right) = 0.
Either t=2t = 2 or t=−2t = - 2
Hence, the roots of the equation t2−4=0{t^2} - 4 = 0 are t=2t = 2 and t=−2t = - 2.
Now, we are given the condition that t>0t > 0. So, t has to be a positive number. So, t=2t = 2 is the correct answer. Hence, option (B) is the right answer.

Note: We should remember the algebraic identities such as a2−b2=(a−b)(a+b){a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right) so as simplify the factorization process of the equations. We must take care of the calculations while solving quadratic equations. We should keep in mind the conditions given to us in the question such as t>0t > 0 so as to get to the correct answer.