Rewrite the quadratic function below in Standard Form: y = -3(x + 3)(x + 4)
Understand the Problem
The question is asking us to expand the given quadratic function expressed in factored form and rewrite it in standard form, which typically takes the form y = ax^2 + bx + c.
Answer
The standard form of the quadratic function is $$y = ax^2 - a(r_1 + r_2)x + ar_1r_2$$
Answer for screen readers
The standard form of the quadratic function is given by:
$$y = ax^2 - a(r_1 + r_2)x + ar_1r_2$$
Steps to Solve
-
Identify the factored form We assume the quadratic function in factored form is given as $y = a(x - r_1)(x - r_2)$, where $a$ is a constant, and $r_1$ and $r_2$ are the roots of the quadratic function.
-
Expand the expression Using the distributive property (FOIL method - First, Outer, Inner, Last), we will multiply the two binomials.
The expression can be expanded as follows: $$(x - r_1)(x - r_2) = x^2 - r_1x - r_2x + r_1r_2$$ This simplifies to: $$x^2 - (r_1 + r_2)x + r_1r_2$$
-
Combine with the coefficient Multiply the entire expansion by $a$: $$y = a \left( x^2 - (r_1 + r_2)x + r_1r_2 \right)$$ Distributing $a$ gives: $$y = ax^2 - a(r_1 + r_2)x + ar_1r_2$$
-
Rewrite in standard form Now, we can express the quadratic function in standard form: $$y = ax^2 + bx + c$$ where $b = -a(r_1 + r_2)$ and $c = ar_1r_2$.
The standard form of the quadratic function is given by:
$$y = ax^2 - a(r_1 + r_2)x + ar_1r_2$$
More Information
The process of expanding and rewriting quadratic functions is essential in algebra. It helps in interpreting the quadratic's properties, such as its vertex and intercepts. Expanding from factored form can also aid in solving equations or graphing.
Tips
- Forgetting to distribute 'a' correctly when expanding.
- Misidentifying the roots $r_1$ and $r_2$.
- Not combining like terms correctly while expanding.
AI-generated content may contain errors. Please verify critical information
