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Quadratic Quadratic Equations In One Equations In One Variable Variable Xandro Alexi Nieto, M.Math Xandro Alexi Nieto, M.Math Educ. Educ. UST – Faculty of Pharmacy UST – Faculty of Pharmacy

Quadratic Equations in One Variable

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Page 1: Quadratic Equations in One Variable

Quadratic Equations In Quadratic Equations In One VariableOne Variable

Xandro Alexi Nieto, M.Math Educ.Xandro Alexi Nieto, M.Math Educ.UST – Faculty of PharmacyUST – Faculty of Pharmacy

Page 2: Quadratic Equations in One Variable

Quadratic Equations in One VariableQuadratic Equations in One Variable

Definition:Definition: an equation that can be written in an equation that can be written in

the form,the form,

aaxx22 + + bbx + x + cc = = 00

where where aa, , bb, and , and cc are integers are integers ((aa,,bb,,cc Z Z ,, a a 0 0 ).).

Page 3: Quadratic Equations in One Variable

Quadratic Equations in One VariableQuadratic Equations in One Variable

A quadratic equation is said to be in A quadratic equation is said to be in standard formstandard form if it is written in if it is written in

aaxx22 + + bbx + x + cc = 0 form. = 0 form.(all the nonzero terms on the left side and (all the nonzero terms on the left side and 0 on the right side of the equation)0 on the right side of the equation)

is said to be in standard form, where a = 5, b = 7 and c = –3.

is not yet in standard form. However, equivalently

is in standard form, where

a = 1, b = – 2 and c = –1.

Page 4: Quadratic Equations in One Variable

Methods of Solving Quadratic Methods of Solving Quadratic Equations in One VariableEquations in One Variable

There are various ways of There are various ways of finding the solution set to finding the solution set to quadratic equations in one quadratic equations in one variable, namely:variable, namely:

•solution by extracting square roots.

•solution by factoring.

•solution by completing the square.

•solution by quadratic formula.

Page 5: Quadratic Equations in One Variable

SOLUTION BY SOLUTION BY

EXTRACTING THE EXTRACTING THE SQUARE ROOTSSQUARE ROOTS

Page 6: Quadratic Equations in One Variable

Solution by Extracting the Solution by Extracting the Square RootsSquare Roots Applicable if the standard form of the Applicable if the standard form of the

given equation given equation

aaxx22 + + bbx + x + cc = 0 = 0

has has bb = 0. = 0.

That is, the equationThat is, the equation a axx22 + + bbx + x + cc = 0 = 0 becomesbecomes aaxx22 + + cc = 0. = 0.

Thus, to solve such form of equations, Thus, to solve such form of equations,

aaxx22 + + cc = 0 = 0

aaxx22 = – = – cc

xx22 = – = – c c // a a

Page 7: Quadratic Equations in One Variable

Solution by Extracting the Solution by Extracting the Square RootsSquare Roots Examples: Solve the following,Examples: Solve the following,

Page 8: Quadratic Equations in One Variable

Solution by Extracting the Solution by Extracting the Square RootsSquare Roots Examples: Solve the following,Examples: Solve the following,

Thus, the solution set of the given equation is

To verify, let’s check the answers (x = ±4) by substituting it to the given equation,

if x = 4 if x = - 4

2(4)2 – 32 2(- 4)2 - 32

2(16) – 32 2(16) – 32

32 – 32 32 – 32

0 0

Page 9: Quadratic Equations in One Variable

Solution by Extracting the Solution by Extracting the Square RootsSquare Roots Examples: Solve the following,Examples: Solve the following,

Page 10: Quadratic Equations in One Variable

Solution by Extracting the Solution by Extracting the Square RootsSquare Roots Examples: Solve the following,Examples: Solve the following,

Try this one on your own, and click me if you already have an answer.

Do you already have an answer?

Click anywhere..

Hope you got

Page 11: Quadratic Equations in One Variable

SOLUTION BY SOLUTION BY

FACTORINGFACTORING

Page 12: Quadratic Equations in One Variable

Solution by FactoringSolution by Factoring

Of course, this method is applicable if Of course, this method is applicable if the given is factorable.the given is factorable.

If it is factorable, then apply the If it is factorable, then apply the properties of real numbers, which properties of real numbers, which states thatstates that

“If r and s are real numbers,

then rs = 0 if and only if

r = 0 or s = 0.”

Page 13: Quadratic Equations in One Variable

Solution by FactoringSolution by Factoring

Examples: Solve the following,Examples: Solve the following,

Factor.

Equate both factors to zero.

Solve the equations separately.

Verify.

Express the answer(s) in solution set.

Page 14: Quadratic Equations in One Variable

Solution by FactoringSolution by Factoring

Examples: Solve the following,Examples: Solve the following,

Express in standard form first!

ax2 + bx + c = 0

Page 15: Quadratic Equations in One Variable

Solution by FactoringSolution by Factoring

Examples: Solve the following,Examples: Solve the following,

Page 16: Quadratic Equations in One Variable

Solution by FactoringSolution by Factoring

Examples: Solve the following,Examples: Solve the following,

Try your best to factor the left side.

Group two terms by two, like

(ax2 – a2x) + (bx – ab) = 0

Now do the rest….

I hope you got this answer,

Page 17: Quadratic Equations in One Variable

SOLUTION BY SOLUTION BY

COMPLETING THE COMPLETING THE SQUARESQUARE

Page 18: Quadratic Equations in One Variable

Solution by Solution by Completing the Square Completing the Square applicable even if the given is NOT applicable even if the given is NOT

factorablefactorableTo perform solution by completing the square, express first the given quadratic equation in standard form.

Transpose the constant c on the right side of the equation.

Divide every term by the numerical coefficient of x2, represented by a.

Complete the square on the left side by adding a constant, which is SQUARE OF HALF

THE COEFFICIENT OF x.

Same constant will be added on the right side of the equation.

Page 19: Quadratic Equations in One Variable

Solution by Solution by Completing the Square Completing the Square

Simplify….

Factor the left side of the equation as the left side always become a perfect square trinomial. Then combine the fractions on the right-side by getting the LCD.

To solve for x, get the square root of the left and right side of the equation.

Leave the variable x alone on the left side of the equation by transposing the other term on the right.

Page 20: Quadratic Equations in One Variable

Solution by Solution by Completing the Square Completing the Square

known as the quadratic formula.

“I will not give you examples on solving quadratic equations by completing the square since most students find it tedious and tiresome. Rather, we are going to make use of quadratic formula for finding solutions to quadratic equations, factorable and NOT factorable.”

Page 21: Quadratic Equations in One Variable

SOLUTION BY SOLUTION BY

QUADRATIC QUADRATIC FORMULAFORMULA

Page 22: Quadratic Equations in One Variable

Solution by the Solution by the Quadratic Formula Quadratic Formula as derived from solution by as derived from solution by

completing the square, any equation completing the square, any equation in the form in the form

has .has .

Page 23: Quadratic Equations in One Variable

Solution by the Solution by the Quadratic Formula Quadratic Formula Let’s take the previous example,Let’s take the previous example,

This is the same example you have seen on slide 7 of this Powerpoint presentation, wherein our answer was ssx: {4,–4}

From the standard form of quadratic equation ax2 + bx + c = 0, 2x2 – 32 = 0 has a = 2, b = 0 and c = - 32.

Thus, our solution set ssx: {4, -4}

Page 24: Quadratic Equations in One Variable

Solution by the Solution by the Quadratic Formula Quadratic Formula Let’s take another previous example,Let’s take another previous example,

This is the same example you have seen on slide 14 of this Powerpoint presentation, wherein our answer was

From the standard form of quadratic equation ax2 + bx + c = 0, 6x2 – 11x – 10 = 0 has a = 6, b = – 11 and c = - 10.

Page 25: Quadratic Equations in One Variable

Thus, the solution set is

Solution by the Solution by the Quadratic Formula Quadratic Formula Let’s take another example,Let’s take another example,

Substitute the to the quadratic formula, a = 2, b = 4, & c = -3.

Page 26: Quadratic Equations in One Variable

Thus, the solution set is

Solution by the Solution by the Quadratic Formula Quadratic Formula Let’s take another example,Let’s take another example,

Substitute the to the quadratic formula, a = 3, b = – 4, & c = 8.

Page 27: Quadratic Equations in One Variable

In general,In general,

when using quadratic formula, the when using quadratic formula, the discriminantdiscriminant gives you a clue of the gives you a clue of the nature of the solutions.nature of the solutions.

The discriminant is the value inside the square root of the quadratic formula.

If b2 – 4ac > 0, then the quadratic equation has 2 distinct real roots (meaning, 2 real-number elements in the solution set).

If b2 – 4ac = 0, then the quadratic equation has 1 real root.

If b2 – 4ac < 0, then the quadratic equation has 2 distinct non-real roots (meaning, 2 imaginary elements, or 2 complex (not real) elements in the solution set).

Page 28: Quadratic Equations in One Variable

AssignmentAssignment

A 10-point online assignment is provided for this topic to assess your learning from this presentation.

Simply click the “Quadratic Equations” of the Assignments panel.