Question

Solve the equation using any method. Explain your reasoning. \(t^2+8 t+16=0\)

Short answer

The solution to the equation \(t^2+8t+16=0\) is \(t=-4\).

Step-by-step solution

Identify the Trinomial

The given equation is a trinomial \(t^2+8t+16\). This particular trinomial is a perfect square, which means it can be factored into squares of binomials, thus simplifying the equation.

Factoring the Trinomial

The equation can be factored into \((t+4)^2=0\), where \(t+4\) is the square root of the trinomial.

Solving the Factored Equation

Now that we have factored the trinomial into \((t+4)^2=0\), we can solve it. Here's how: if \((t+4)^2=0\), then \(t+4 = 0\) because if you square zero, you get zero, and if you square any non-zero number, you get a positive number.

Final Solution

Finally, when \(t+4 = 0\), solving it results in \(t = -4\) as the final solution.

Key concepts

Factoring Trinomials

Factoring trinomials can seem tricky at first, but once you understand the process, it becomes much more manageable. A trinomial is a type of polynomial that has three terms. When we factor a trinomial, the goal is to rewrite it as a product of simpler polynomials, usually binomials. Consider the trinomial \(t^2 + 8t + 16\). Notice how the trinomial is structured in the form \(ax^2 + bx + c\). Here, \(a=1\), \(b=8\), and \(c=16\).

To factor, we look for two numbers that multiply to \(c\) (16 in this case) and add up to \(b\) (8). These two numbers are 4 and 4. We can then express the original trinomial as the square of a binomial: \((t+4)(t+4)\), which simplifies to \((t+4)^2\). Factoring transforms the equation into a simpler form, making solving much easier.

Perfect Square Trinomials

A perfect square trinomial is an exceptional case where a trinomial can be written as the square of a binomial. Recognizing a perfect square trinomial can save you time and simplify your calculations. These appear in the format \((a+b)^2\) or \((a-b)^2\), which expands to \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\).

In the example equation \(t^2 + 8t + 16\), we see it follows the structure of \((a+b)^2\), where both middle term \(8t\) and the constant 16 correspond to \(2ab\) and \(b^2\) respectively. Here, both \(a\) and \(b\) are 4, fitting the form \((t+4)^2\). Knowing this, we simplify the equation to \((t+4)^2 = 0\) with confidence, understanding that our trinomial is indeed a perfect square.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. These equations often appear in the form \(ax^2 + bx + c = 0\). Once you factor the quadratic equation, solving it becomes straightforward.

Given the equation \((t+4)^2 = 0\), you've already factored the quadratic. From here, you'll set the factored expression \((t+4)\) equal to zero as part of the zero-product property. This property tells us that if the product of multiple terms is zero, at least one of those terms must be zero. Thus, we solve \(t+4 = 0\), and find that \(t = -4\).

• Check your solution: Substitute back into the original equation to verify.
• Remember: Factoring is a strategy. If it doesn’t work, try other methods like completing the square or using the quadratic formula.

By following these steps, solving quadratic equations becomes a less daunting task.