Factor Quadratics - International Brotherhood of Electrical Workers Exam IBEW

1. What is the factoring method called when you factor $ax^2+bx+c$ by splitting the middle term?

   Factor by grouping (split-the-middle-term method). Splitting the middle term allows grouping into pairs that factor to common binomials.

2. Identify the factored form of $x^2+10x+25$.

   $(x+5)^2$. The trinomial is a perfect square with middle term twice the product of x and 5.

3. Identify the factored form of $x^2-9x+20$.

   $(x-5)(x-4)$. The constants -5 and -4 sum to -9 and multiply to 20, matching the trinomial coefficients.

4. What is the factoring pattern for a common binomial factor: $ab+ac$?

   $a(b+c)$. Factoring out the common factor $a$ groups the remaining terms into a binomial.

5. Identify the factored form of $3x^2-11x-4$.

   $(3x+1)(x-4)$. The AC method yields factors with cross terms summing to -11x and constant -4.

6. Identify the factored form of $2x^2+7x+3$.

   $(2x+1)(x+3)$. Using the AC method, factors are found where cross terms sum to 7x and product is 3.

7. What is the AC test condition used to factor $ax^2+bx+c$ by grouping?

   Find $m,n$ with $mn=ac$ and $m+n=b$. The AC method identifies numbers that multiply to $ac$ and add to $b$ for splitting the middle term.

8. Identify the factored form of $9x^2-25$.

   $(3x-5)(3x+5)$. This difference of squares has square roots 3x and 5, forming conjugate factors.

9. Find and correct the error: $x^2-6x+9=(x-3)(x+3)$.

   Correct: $x^2-6x+9=(x-3)^2$. The error misapplies difference of squares instead of recognizing the perfect square trinomial.

10. What is the factoring pattern for a difference of squares: $a^2-b^2$?

    $(a-b)(a+b)$. The difference of squares formula states that $a^2 - b^2$ factors into the product of the sum and difference of $a$ and $b$.

11. What is the factoring pattern for a perfect square trinomial: $a^2-2ab+b^2$?

    $(a-b)^2$. A perfect square trinomial with negative middle term factors as the square of the binomial difference.

12. What is the standard form of a quadratic expression in $x$?

    $ax^2+bx+c$ with $a\ne 0$. The standard quadratic form has a non-zero leading coefficient for the squared term.

13. What is the factoring pattern for a four-term expression: $ax+ay+bx+by$?

    $(a+b)(x+y)$. Grouping terms pairs the common factors to yield two binomials.

14. What is the factoring pattern for a trinomial with leading coefficient $1$: $x^2+bx+c$?

    $(x+m)(x+n)$ where $m+n=b$ and $mn=c$. For quadratics with leading coefficient 1, factors are binomials whose constants sum to $b$ and multiply to $c$.

15. Identify the factored form of $x^2-16$.

    $(x-4)(x+4)$. This is a difference of squares where $x^2 - 4^2$ factors into conjugates.

16. Identify the factored form of $x^2-12x+36$.

    $(x-6)^2$. The trinomial matches the perfect square pattern with middle term negative twice x times 6.

17. Identify the factored form of $6x^2+11x+3$.

    $(3x+1)(2x+3)$. Splitting yields groups that factor to binomials with cross terms adding to 11x.

18. Identify the factored form of $x^2+2x-15$.

    $(x+5)(x-3)$. The constants 5 and -3 sum to 2 and multiply to -15, aligning with the trinomial terms.

19. Identify the factored form of $4x^2-12x+9$.

    $(2x-3)^2$. The trinomial is a perfect square of 2x - 3, as the middle term is twice their product.

20. What is the first step when factoring $ax^2+bx+c$ if all terms share a common factor?

    Factor out the GCF first. Factoring out the greatest common factor simplifies the quadratic before applying other methods.

21. Identify the factored form of $x^2+7x+12$.

    $(x+3)(x+4)$. The constants 3 and 4 sum to 7 and multiply to 12, fitting the trinomial pattern.

22. Identify the factored form of $5x^2+5x$.

    $5x(x+1)$. Factoring out the GCF 5x leaves a binomial factor.

23. What is the factoring pattern for a perfect square trinomial: $a^2+2ab+b^2$?

    $(a+b)^2$. A perfect square trinomial with positive middle term factors as the square of the binomial sum.

24. Identify the factored form of $2x^2-8x$.

    $2x(x-4)$. The GCF 2x factors out, leaving a linear binomial.

25. Identify the factored form of $x^2+x-12$.

    $(x+4)(x-3)$. Constants 4 and -3 sum to 1 and multiply to -12, fitting the trinomial.

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