How to convert factored form to standard form?
Understand the Problem
The question is asking for the method or steps involved in converting a quadratic equation from factored form (such as (x - p)(x - q)) to standard form (usually ax^2 + bx + c). This involves expanding the factored expression and combining like terms.
Answer
The standard form is $x^2 + (-p - q)x + pq$.
Answer for screen readers
The standard form of the quadratic equation obtained from the factored form $(x - p)(x - q)$ is: $$ x^2 + (-p - q)x + pq $$
Steps to Solve
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Identify the factored form Recognize the factored form of the quadratic equation, which is given as $(x - p)(x - q)$.
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Distribute the terms Use the distributive property (also known as the FOIL method for binomials) to expand the expression:
- Multiply $x$ by both terms in the second binomial: $$ x \cdot (x - q) = x^2 - qx $$
- Then, multiply $-p$ by both terms in the second binomial: $$ -p \cdot (x - q) = -px + pq $$
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Combine like terms Now, combine the results from the distribution: $$ x^2 - qx - px + pq $$
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Simplify the expression Combine the coefficients of $x$: $$ x^2 + (-q - p)x + pq $$
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Write in standard form The final expression is now in standard form $ax^2 + bx + c$, where:
- $a = 1$
- $b = -p - q$
- $c = pq$
The standard form of the quadratic equation obtained from the factored form $(x - p)(x - q)$ is: $$ x^2 + (-p - q)x + pq $$
More Information
Converting from factored form to standard form helps in understanding the overall properties of the quadratic function, including finding the vertex, axis of symmetry, and other characteristics like the direction the parabola opens.
Tips
- Forgetting to distribute both terms correctly can lead to incorrect coefficients. Make sure to multiply the entirety of both binomials.
- Failing to combine like terms properly could result in inconsistent forms. Always check that all coefficients are accurately added together.
