Problem 7 Factor the expression. $$ t^... [FREE SOLUTION]

Chapter 10: Problem 7

Factor the expression. $$ t^{2}+10 t+25 $$

Short Answer

Expert verified

The factored form of the expression $t^{2}+10t+25$ is $(t + 5)^2$.

Step by step solution

Identify the form

First, recognize that the given expression follows the form of a perfect square trinomial which is $(ax)^2 + 2abx + b^2$. Here, $a = t$, $b = 5$ and 2ab = 10.

Factor the expression

When the quadratic takes the form of a perfect square trinomial, you can simplify it to $(ax + b)^2$. Therefore, $t^{2}+10t+25$ can be factored as $(t + 5)^2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be made by squaring a binomial.
They take the form $(ax)^2 + 2abx + b^2$.
The key features are:

The first and last terms are perfect squares.
The middle term is twice the product of the terms that are squared in the first and last position.

For example, in the expression $t^2 + 10t + 25$,
we see that the

first term $t^2$ is a square of $t$,
last term $25$ is a square of 5,
and the middle term $10t$ is exactly $2 \times t \times 5$.

This pattern confirms it's a perfect square trinomial.

Quadratic Expression

A quadratic expression is an algebraic expression of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
In this form, $x$ is the variable, and the highest power of $x$ is 2.
Quadratics are quite common in algebra and appear in different guises, such as factored form or standard form.Quadratics can often be simplified or rewritten into another form.
When it comes to identifying a quadratic expression as a perfect square trinomial, recognize the pattern mentioned earlier.
This will help in rewriting the expression efficiently.

Standard form: $ax^2 + bx + c$
Factored form: $(ax + b)^2$ for perfect square trinomials

Step by Step Solution

When tasked with factoring a quadratic expression, approaching the problem step-by-step can be tremendously beneficial.
Here, we break down the tasks necessary to factor the expression $t^2 + 10t + 25$.Start by identifying if the given expression is a perfect square trinomial:

In our example, we found that $(t)^2$ aligns with the first term, and $(5)^2$ matches the last term.
The middle term $10t$ confirms the pattern $2 \times t \times 5$.

Next, factor the expression:
To factor, use the form $(ax + b)^2$.
Given $t^2 + 10t + 25$, it simplifies to $(t + 5)^2$.
This factoring shows the expression can be rewritten compactly and helps solve equations involving the quadratic.
Always check your work to confirm that multiplying back results in the original expression.

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91影视

Factor the expression. $$ t^{2}+10 t+25 $$

Short Answer

Step by step solution

Identify the form

Factor the expression

Key Concepts

Perfect Square Trinomial

Quadratic Expression

Step by Step Solution

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