Solving for the Roots Solving for the Roots of a Quadratic of a Quadratic EquationEquation

12 October 201012 October 2010

What do you know about What do you know about solving quadratic solving quadratic equations?equations?

VocabularyVocabulary

FactorFactor– 2 numbers whose product is the 2 numbers whose product is the

desired numberdesired number– Usually occur in pairsUsually occur in pairs– Example:Example:

Factors of 4 are 1, 4 but also 2Factors of 4 are 1, 4 but also 2

– Finding factors on your calculatorFinding factors on your calculator– Your turn:Your turn:

Find all possible the factors of 144 and Find all possible the factors of 144 and 256256

Vocabulary, cont.Vocabulary, cont.

FactoringFactoring– The process of rewriting a number or The process of rewriting a number or

an expression at the product of its an expression at the product of its factorsfactors

– Examples:Examples: 4 = (1)(4)4 = (1)(4) 4 = (2)(2)4 = (2)(2)

Vocabulary, cont.Vocabulary, cont.

Standard Quadratic FormStandard Quadratic Form

f(x) = axf(x) = ax22 + bx + c = 0, a + bx + c = 0, a ≠ ≠ 00

Leading

Coefficient

Constant

Why factor?Why factor?

Factoring allows us to solve for Factoring allows us to solve for the x-values of a quadratic the x-values of a quadratic equation.equation.

We can find either the roots of We can find either the roots of the equation or the value of x for the equation or the value of x for a specific value of y.a specific value of y.

By solving for the roots, we can By solving for the roots, we can graph a quadratic equation more graph a quadratic equation more easily by hand.easily by hand.

Preparing to FactorPreparing to Factor

Make sure your equation is in Make sure your equation is in standard quadratic form!standard quadratic form!

Make a list of possible factors Make a list of possible factors (The list can be a mental list or on (The list can be a mental list or on your calculator.)your calculator.)

Method 1: Greatest Method 1: Greatest Common Factor (GCF)Common Factor (GCF) When we use it:When we use it:

– When all the terms share When all the terms share at least at least oneone common factor (The terms can common factor (The terms can share more than one common share more than one common factor.)factor.)

– Generally, there Generally, there isn’tisn’t a constant a constant term.term.

Method 1: Steps for Method 1: Steps for GCFGCF Step 1: List the common factors of Step 1: List the common factors of

each of the terms in the each of the terms in the equation/expression.equation/expression.

Step 2: Factor out the common factors.Step 2: Factor out the common factors. Step 3 (if solving for the roots): Set the Step 3 (if solving for the roots): Set the

equation/expression equal to 0.equation/expression equal to 0. Step 4: Solve for x by separating into Step 4: Solve for x by separating into

two different equations equal to 0.two different equations equal to 0. Step 4: Discard any mathematically Step 4: Discard any mathematically

impossible answers, ex. 4 = 0.impossible answers, ex. 4 = 0.

Method 1: GCF Method 1: GCF ExamplesExamples Solve for the roots of the Solve for the roots of the

following equations:following equations:1.1. y = -xy = -x22 + 6x + 6x

Method 1: GCF Method 1: GCF Examples, cont.Examples, cont.

1.1. y = 21xy = 21x55 + 49x + 49x33

Method 2: Difference Method 2: Difference of Squaresof Squares When we use it:When we use it:

– When the leading coefficient term When the leading coefficient term (ax(ax22) and the constant term (c) are ) and the constant term (c) are both perfect squares.both perfect squares.

– ANDAND there is no middle term (bx). there is no middle term (bx).

Method 2: Steps for Method 2: Steps for Difference of SquaresDifference of Squares Step 1: Find the square root of the Step 1: Find the square root of the

leading coefficient term (axleading coefficient term (ax22).). Step 2: Find the square root of the Step 2: Find the square root of the

constant term (c).constant term (c). Step 3: Write two sets of parentheses. Step 3: Write two sets of parentheses.

Write the square roots of each term inside Write the square roots of each term inside the parentheses.the parentheses.

Step 4: In the 1Step 4: In the 1stst set of parentheses, set of parentheses, write + in between the two terms. In the write + in between the two terms. In the 22ndnd set of parentheses, write — in set of parentheses, write — in between the two terms.between the two terms.

Step 5: Set both sets of parentheses Step 5: Set both sets of parentheses equal to 0 and solve for x.equal to 0 and solve for x.

Method 2: Examples Method 2: Examples of Difference of of Difference of SquaresSquares Solve for the roots of the Solve for the roots of the

following equations:following equations:

1.1. y = xy = x22 - 25 - 25

Method 2: Examples of Method 2: Examples of Difference of Squares, Difference of Squares, cont.cont.1.1. y = 4xy = 4x22 - 81 - 81

Method 2: Examples of Method 2: Examples of Difference of Squares, Difference of Squares, cont.cont.1.1. y = 9xy = 9x22 - 36 - 36

Method 3: Box MethodMethod 3: Box Method

When we use it:When we use it:– When we can’t use GCF or the When we can’t use GCF or the

difference of squares.difference of squares.

Method 3: Steps in Method 3: Steps in the Box Methodthe Box Method Step 1: List the factors of the leading Step 1: List the factors of the leading

coefficient and the constant.coefficient and the constant. Step 2: Pick a set of factors for both the Step 2: Pick a set of factors for both the

leading coefficient and the constant.leading coefficient and the constant. Step 3: Cross multiply the numbers. Step 3: Cross multiply the numbers.

Check if it’s possible to combine the Check if it’s possible to combine the products to create the middle term (b). products to create the middle term (b).

Step 4: If not, pick a different set of Step 4: If not, pick a different set of factors and try the process again. If yes, factors and try the process again. If yes, then write the equation as the product of then write the equation as the product of those factors.those factors.

Step 5: Set the equation equal to 0 and Step 5: Set the equation equal to 0 and solve for x.solve for x.

Method 3: Examples Method 3: Examples of the Box Methodof the Box Method Solve for the roots of the Solve for the roots of the

following equations:following equations:

1.1. y = xy = x22 + 7x + 6 + 7x + 6

Method 3: Examples Method 3: Examples of the Box Method, of the Box Method, cont.cont.1.1. y = 2xy = 2x22 – 5x + 3 – 5x + 3

Method 3: Examples Method 3: Examples of the Box Methodof the Box Method y = 4xy = 4x22 – 24x + 11 – 24x + 11

Method 4: Quadratic Method 4: Quadratic EquationEquation When we use it:When we use it:

– When an equation or expression can’t When an equation or expression can’t be factored.be factored.

a

acbbx

2

42

Method 4: Steps for Method 4: Steps for the Quadratic Equationthe Quadratic Equation Step 1: Make sure that the equation Step 1: Make sure that the equation

is in standard quadratic form!is in standard quadratic form! Step 2: List the values for a, b, and Step 2: List the values for a, b, and

c.c. Step 3: Substitute the appropriate Step 3: Substitute the appropriate

values for a, b, and c into the values for a, b, and c into the formula for the quadratic equation.formula for the quadratic equation.

Step 4: Solve for x. Step 4: Solve for x. Remember Remember that you will have two solutions!that you will have two solutions!

Method 4: Examples Method 4: Examples using the Quadratic using the Quadratic EqnEqn Solve for the roots of the Solve for the roots of the

following equations:following equations:

1.1. xx22 -12x + 4 = 0 -12x + 4 = 0

Method 4: Examples Method 4: Examples using the Quadratic Eqn, using the Quadratic Eqn, cont.cont.1.1. xx22 + 6x + 3 = 0 + 6x + 3 = 0

Method 4: Examples Method 4: Examples using the Quadratic Eqn, using the Quadratic Eqn, cont.cont.1.1. 2x2x22 – 7x – 5 = 0 – 7x – 5 = 0