Mathematics
Factoring Quadratic Equations
Factoring quadratic equations involves finding the two binomial factors of a quadratic expression. This process helps to solve quadratic equations and graph parabolas. By factoring, we can easily identify the roots or x-intercepts of the quadratic equation, which are the points where the parabola intersects the x-axis. This method is a fundamental skill in algebra and calculus.
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- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
And the point is that, if we now have a polynomial divided by anything, it had better divide evenly into every term of the polynomial or all (or most) bets are off. For example,because –5 divides evenly into each term in the numerator. If the coefficient of the x3 term had been a 7, then we wouldn’t have been able to do this division so smoothly. By the way, the reason we say most (and not all) bets are off is that it can be done, but we’ll leave that to the mathematicians.Just as factoring was important when dividing any two quantities, whether they are numbers, fractions, or anything else, factoring is an important part of dividing two polynomials. For example, by recognizing that there is a common factor (x) in the numerator and denominator of we getIn fact, in this example, the numerator can be factored further, and the whole expression equals (x + 5). But we are getting ahead of ourselves a little here. First we must talk about quadratic equations and how to factor them, the topic of the next section.Quadratic Equations
A quadratic equation is an equation in which the unknown is squared and there is no higher power of the unknown. It is okay if there are no lower powers of the unknown (in other words, no “x” term or no “pure number” term). An example of a quadratic equation is x2 +x –6 =0, and so are x2 – 9 = 0 and x2 +3x = 0. Quadratic equations always have two answers for the value of the unknown (even though at times they are the same number twice).The general form of a quadratic equation is ax2 +bx +c = 0, where b and c can be any numbers, even 0, as we saw with the examples of x2 –9 = 0 and x2 +3x = 0. If a = 0, though, we no longer have a quadratic–according to the definition, there has to be a squared term.Solving a Quadratic Equation
The solutions to a quadratic equation are based on a simple fact: if two factors are multiplied together and the product is 0, then either one or both of the factors must equal 0. There just are no two nonzero numbers whose product is 0. Period. We always write a quadratic equation on one side of the equals sign with 0 on the other so we can see what the factors are.- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
To factor a quadratic trinomial into the product of two binomials, use the reverse of double distribution.FACTORING x2 + bx + c
Since (x + 2)(x + 5) = x2 + 7x + 10, you know that x2 + 7x + 10 = (x + 2)(x + 5). Notice that the binomial factors contain 2 and 5 since these are the only two integers that, when multiplied together, give 10, the last term in x2 + 7x + 10, and, when added together, equal 7, the coefficient of the x-term in x2 + 7x + 10. If you start with x2 + 7x + 10 and want to know its factors, work in reverse:1.Write the general form of the binomial factors:2.Fill in the missing terms in the binomial factors with the two numbers whose product is the last number term of the quadratic trinomial and whose sum is the numerical coefficient of the middle x-term . Since 2 · 5 = 10 and 2 + 5 = 7:Always check your work by multiplying the two binomial factors together and then comparing the product to the original quadratic trinomial. EXERCISE 1 Factoring a Quadratic TrinomialFactor y2 − 7y + 12 as the product of two binomials.SOLUTIONFind the two numbers whose product is +12 and whose sum is –7. Because the product of the two numbers you are seeking is positive (+12) and their sum is negative (−7), both numbers must be negative. Since(–3) · (–4) = +12 and (–3) + (–4) = –7 the correct factors of +12 are −3 and −4. Hence:y2 – 7y + 12 = (y – 3)(y – 4)EXERCISE 2 Factoring a Quadratic TrinomialFactor n2 − 5n − 14 as the product of two binomials.SOLUTIONFind the two numbers whose product is −14 and whose sum is −5. The two factors of −14 must have opposite signs. Since(+2) · (–7) = –14 and (+2) + (–7) = –5, the correct factors of −14 are +2 and −7. Hence:n2 – 5n – 14 = (n + 2)(n – 7)FACTORING ax2 + bx + c (a > 1)
As you can imagine, factoring a quadratic trinomial becomes more difficult when the numerical coefficient of the x2 -term is different from 1 since there are more possibilities to consider. To factor 3x2 + 10x- eBook - ePub
Differentiating Instruction in Algebra 1
Ready-to-Use Activities for All Students (Grades 7-10)
- Kelli Jurek(Author)
- 2021(Publication Date)
- Routledge(Publisher)
Directions: In your groups, each student will take turns rolling the die. Answer the corresponding question on the record sheet. You may discuss your answers with your group members before recording your answer. You must answer all of the questions.Name:────────── Hour/Block:───── Date:─────Lesson 3: Solving Using the Quadratic Formula
Think Dots
Directions: In your groups, each student will take turns rolling the die. Answer the corresponding question on the record sheet. You may discuss your answers with your group members before recording your answer. You must answer all of the questions.Name:────────── Hour/Block:───── Date:─────Lesson 3: Solving Using the Quadratic Formula
Think Dots Record Sheet
Circle worksheet chosen:Name:────────── Hour/Block:───── Date:─────Lesson 3: Solving Using the Quadratic Formula
Exit Slip
- Write the quadratic formula.
- Explain why the quadratic formula can be used to solve any quadratic function.
- Explain why there may be two solutions to a real-life quadratic function application, but one answer can sometimes be eliminated.
Lesson 4: Solving by Factoring
Checkerboard Small-Group Activity Lesson Plan
Purpose: Students will work in small groups (in pairs or groups of three) to put together the checkerboard puzzle that reviews identifying solutions to linear equations and identifying the slope and y-intercept from an equation.Prerequisite Knowledge Materials Needed Students should know:• how to find factors of numbers,• how to use FOIL to multiply binomials,• the exponent rules for multiplication,• how to combine like terms, and• the integer multiplication rules. • One copy of the checkerboard for each small group of students, cut into the squares• Scrap paper for solving equations Lesson
- ➤ Teachers may want to use random grouping. Each group should receive a packet of the puzzle pieces that are in no particular order.
- ➤ Students are asked to put together the checkerboard by matching pieces that have questions and answers that go together. There is no particular message revealed when the puzzle is completed.
- ➤ Students may notice that there are puzzle pieces that are blank on one side of the piece. They may figure out that these pieces belong on the left or right border of the puzzle.
- eBook - ePub
GMAT Advanced Quant
250+ Practice Problems & Online Resources
- (Author)
- 2020(Publication Date)
- Manhattan Prep(Publisher)
Quadratic Templates
On the GMAT, quadratic expressions take three common forms called the Quadratic Templates. Memorize these templates, and get comfortable transforming back and forth between factored and distributed form:FactoredDistributedSquare of a Sum (a + b)2= a2 + 2ab + b2Square of a Difference (a − b)2= a2 − 2ab + b2Difference of Two Squares (a + b) (a − b)= a2 − b2Quick Manipulation
Expressions with both squared and non-squared common terms should make you suspect that you are looking at a Quadratic Template.Try-It #6-4
Factor .Once you are comfortable with Quadratic Templates, you can manipulate even complicated expressions quickly, as in the middle box above. Until then, write down the templates and the substitution of the common terms, as in the box on the right.The very same problem could have been presented in disguise:Factor .The common terms are slightly harder to spot in this form. In such a case, start with the squared terms, and 25y2 . Then, try to untangle their square roots, and 5y, from the remaining term. The factored form is:The Middle Term: 2ab
The square of a sum and square of a difference templates have something in common: the middle term is ±2ab. The only difference is the sign of that middle term.When you add these two templates, the middle terms cancel, leaving the end terms:Factored Distributed Square of a Sum (a + b)2= a2 + 2ab + b2+ Square of a Difference (a − b)2= a2 − 2ab + b2Addition: (a + b)2 + (a − b)2= 2a2 + 0 + 2b2 2(a2 + b2 )In contrast, when you subtract these two templates, the end terms cancel, leaving the middle