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

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 ).).

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.

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.

SOLUTION BY SOLUTION BY

EXTRACTING THE EXTRACTING THE SQUARE ROOTSSQUARE ROOTS

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

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

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

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

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

SOLUTION BY SOLUTION BY

FACTORINGFACTORING

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.”

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.

Solution by FactoringSolution by Factoring

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

Express in standard form first!

ax2 + bx + c = 0

Solution by FactoringSolution by Factoring

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

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,

SOLUTION BY SOLUTION BY

COMPLETING THE COMPLETING THE SQUARESQUARE

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.

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.

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.”

SOLUTION BY SOLUTION BY

QUADRATIC QUADRATIC FORMULAFORMULA

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 .

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}

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.

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.

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.

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).

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.