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Quadratic Expressions Mind Map (Factoring (Factoring Quadratic Expressions…
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- Find a pair of numbers that would have the product of c and the sum of b. x^2 + 3 - 18, 6 · (-3) = -18, 6 + (-3) = 3
- Split up the polynomial into 2 different sections. mx + my - 3x + 3y, --> mx + my, -3x + 3y
- Find a pair of numbers that would have the same product of a · c and the sum of b. 3x^2 + 8x + 4, 3 · 4 = 12 --> 2 + 6 = 8, 2 · 6 = 12
- Multiply a by c and d then multiply b by c and d. (12x + 7)(3x - 4) --> 12x · 3x = 36x^2, 12x · (-4) = (-48x), 7 · 3x = 21x, 7 · (-4) = (-28)
- Multiply every term in the bracket by a. 3x(9x - 4), 3x · 9x = 27x^2, 3x · (-4) = (-12x)
Requirements for shortcut:
Must be a binomial, Must be subtraction, Must have perfect square terms
Requirements for shortcut:
Must be trinomial, Must have perfect square temrs for ax and c
- Insert the values of a and b into the expression a^2 + 2ab + b^2 or a^2 - 2ab + b^2 depending on the sign and solve it. (2x + 3)^2, (2x)^2 + 2(2x)(3) + (3)^2 --> 4x^2 + 12x + 9
- Solve the new expression. x^2 - 14^2 --> x^2 - 196
- Find the GCF between the polynomial. 14x^4 + 7x^7, GCF = 7
- Find out if the 2 terms share a variable (Also if the exponents share a power, when taking a variable also take the greatest power). 14x^4 + 7x^7, they share x^4
- Combine the GCF and variable to make a term and divide the expression using your new term. 14x^4/7x^4 + 7x^7/7x^4 --> 2 + x^3
- The dividend is then put in a bracket and multiplied by the GCF and variable.
7x^4(2 + x^3)
- Square root each of the terms in the binomial. 4x^2 - 144 --> √4x^2 - √144 --> 2x - 12
- Place the square rooted terms into 2 brackets multiplying each other and have the subtract themselves in one bracket and add in another. 2x - 12 --> (2x - 12)(2x + 12)
- Square root the terms ax and c.
4x^2 + 20x + 25 --> √4x^2, √25 --> 2x, 5
- Multiply ax by c and then that by 2. 2 · 5 · 2x --> 20x
- If the value of the product is equal to bx in the original expression, put ax and c adding each other in a bracket with the power of 2. 2 · 5 · 2x = 20x, (2 + 5)^2
- Divide the expression by the common binomial in the brackets. 3(x + 7)/(x + 7) + 4(x + y)/(x + 7)
- Place the dividend in a bracket multiplying the binomial you divided by. (3 + 4)(x + 7)
- MCF each of the separated parts. mx + my, -3x - 3y --> m(x + y), -3(x + y)
- Rejoin the seperated sections and see if they can be factored any further. m(x + y), -3(x + y) --> m(x + y) - 3(x + y) --> (m + 3)(x + y)
- Place each number in a bracket multiplying it by x (keep track of the signs). 6, -3 --> (x + 6)(x - 3)
- Decompose the value of b to have the 2 numbers you got replace it but have them keep the same base. 3x^2 + 8x + 4 --> 3x^2 + 2x + 6x + 4
- Group factor your new expression. 3x^2 + 2x + 6x + 4 --> x(3x + 2) + 2(3x + 2)
- See if you can factor it any further using other methods. x(3x + 2) + 2(3x + 2) --> (3x + 2)(x + 2)
If you ever want to check if your factoring work is correct, use FOIL and see if you end up with the same expression you started with, if you did, you did it right.
Before factoring with any other method, ALWAYS see if you can MCF.
- Put all your products into an expression and collect like terms. 36x^2 - 48x + 21x - 28 --> 36^2 - 27x - 28
- Put all of your products into an expression and collect like terms if needed. 27x^2, (-12x) --> 27x^2 - 12x
- Get rid of the brackets and square the a value and square the b value and have them subtract each other. (x - 14)(x + 14) --> x^2 - 14^2