Question

Solve the equation. $$(t-19)^{2}=0$$

Short answer

The solution to the equation \((t-19)^{2}=0\) is \(t=19\).

Step-by-step solution

Analyzing the Quadratic equation

Analyze the given quadratic equation \((t-19)^{2}=0\). This is already a squared term equal to zero which implies that the term inside the bracket, i.e., \(t-19\) is equal to zero.

Solving the equation

Solve the equation by setting \(t-19=0\). To solve this, add 19 to both sides of the equation to isolate \(t\). Doing this results to \(t=19\).

Checking the solution

Check your answer by substituting \(t=19\) into the original equation. The equation becomes \((19-19)^{2}=0\), that reduces to \(0^{2}=0\). Since both sides match, our solution is correct.

Key concepts

Quadratic Equation

Quadratic equations are a type of polynomial equation that take the general form of
\[ ax^2 + bx + c = 0 \]
with \( a \), \( b \), and \( c \) being constants, and \( a \) not equal to zero since that would render the equation linear rather than quadratic. In our exercise, the given equation takes the shape of
\[ (t-19)^2 = 0 \]
which is a quadratic equation with imaginary values of \( a = 1 \), \( b = -38 \), and \( c = 361 \). However, it's been simplified to a perfect square set equal to zero. This compact form is a clue that our solution will be straightforward, leading to a single value for \( t \) that makes the equation true.

Algebraic Operations

Algebraic operations include processes like addition, subtraction, multiplication, division, and the rules of exponentiation used to manipulate algebraic expressions and solve equations. When solving for \( t \) in the given quadratic equation, we're demonstrating a basic but crucial algebraic operation: isolating the variable. By adding 19 to both sides of the equation \( t-19 = 0 \), we use the additive property of equality to keep the equation balanced while moving the constant term to the other side, simplifying to \( t = 19 \).
These operations are the fundamental tools we use to solve equations and better understand the relationships between numbers and symbols in algebra.

Solution Verification

Solution verification is vital in mathematics to confirm that our proposed answer indeed satisfies the original equation. By substituting the value of \( t = 19 \) back into the equation and simplifying, we should arrive at a true statement if the solution is correct. If we plug \( t = 19 \) into our quadratic equation, we get
\[ (19-19)^2 = 0 \]
which simplifies to \( 0^2 = 0 \), a true statement. This means \( t = 19 \) is the correct solution to the equation. Verification ensures that any errors made in the algebraic process are caught and that our answer is mathematically sound.