Factor completely. 9q^2+6q+1

To factor the quadratic expression \(9q^ 2 + 6q + 1\) completely, we look for factors of the form \(( aq + b) ^ 2\).

1. Identify the coefficients: \(a = 9\), \(b = 6\), and \(c = 1\).
2. Notice that the expression \(9q^ 2 + 6q + 1\) is a perfect square trinomial.
3. Rewrite the expression as \(( 3q + 1) ^ 2\).

Supplemental Knowledge

Factoring quadratic expressions is a fundamental skill in algebra. A quadratic expression is typically in the form \(ax^ 2 + bx + c\). To factor it completely, you need to find two binomials that multiply to give the original quadratic expression.

1. Standard Form:
   • The standard form of a quadratic expression is \(ax^ 2 + bx + c\).
2. Factoring Process:
   • Look for two numbers that multiply to \(ac\) (the product of the coefficient of \(x^ 2\) and the constant term) and add up to \(b\) (the coefficient of \(x\)).
   • Rewrite the middle term using these two numbers and then factor by grouping.

Concepts to Actions

Learning how to factor quadratics is invaluable when faced with everyday situations that necessitate factored quadratic equations - for instance physics problems related to projectile motion where its path can be represented using quadratic equations; solving factored quadratics allows one to predict when an object will hit the ground via factored quadratic equation.
Imagine you're designing a garden and want to incorporate a parabolic archway. At each point in its height can be described by a quadratic equation; factoring this equation allows you to pinpoint key locations along its arch and ensure both functionality and esthetic appeal in your design.

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