what is 9×2 – 30x + 25?
The Correct Answer and Explanation is:
The expression 9x² – 30x + 25 is a quadratic trinomial. To simplify or factor this expression, let us try factoring it.
We are given:
9x² – 30x + 25
This trinomial is in the standard quadratic form ax² + bx + c, where:
- a = 9
- b = -30
- c = 25
We need to check if this quadratic expression is a perfect square trinomial. A perfect square trinomial takes the form:
(mx – n)² = m²x² – 2mnx + n²
Let’s try:
(3x – 5)²
Now expand this to see if it matches the original expression:
(3x – 5)(3x – 5)
= 3x(3x) – 3x(5) – 5(3x) + 5(5)
= 9x² – 15x – 15x + 25
= 9x² – 30x + 25
Yes, it matches exactly.
Final Answer:
9x² – 30x + 25 = (3x – 5)²
Explanation
To understand the expression 9x² – 30x + 25, it helps to identify its structure. This is a quadratic trinomial, which means it has three terms and the highest exponent is two. The goal is often to factor it to find its roots or to simplify it for solving equations.
Factoring involves expressing a quadratic trinomial as the product of two binomials. One method is checking whether the trinomial is a perfect square. This is a special case where both the first and last terms are perfect squares, and the middle term is twice the product of their square roots.
In 9x² – 30x + 25, the first term, 9x², is a perfect square because it is the square of 3x. The last term, 25, is a perfect square because it is the square of 5. Now check the middle term: 2 × 3x × 5 = 30x. Since the middle term is -30x, it matches -2ab from the square of a binomial. This confirms that the expression is a perfect square trinomial.
So, we can write the expression as:
(3x – 5)²
Factoring in this way helps solve quadratic equations, analyze graphs, and understand the behavior of functions. Recognizing perfect squares quickly is a useful algebra skill that saves time and simplifies more complex math problems.
