Chapter 1 Quadratic Equations PDF These values of x give

solving-quadratic-equations.pdf

review three different ways to solve a quadratic equation. EXAMPLE 1: Solve: 6 . 2 + − 15 = 0. SOLUTION. We check to see if we can factor and find that 6 .

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Factoring and Solving Quadratic Equations Worksheet. Math Tutorial Lab Special Topic∗. Example Problems. Factor completely. 1. 3x + 36. 2. 4x2 + 16x. 3. x2

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Chapter 1 Quadratic Equations

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Precalculus: Quadratic Equations Practice Problems Questions Include complex solutions in your answers 1 Solve (x+9)2 = 21 2 Solve (4x?3)2 = 36 3 Solve (5x?2)2 ?25 = 0 4 Solve by completing the square x2 +6x+2 = 0 5 Solve by completing the square x2 ?14x = ?48 6 Solve by completing the square x2 3 ? x 3 = 3 7

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What is the difference between linear and quadratic equations?

A linear equation is where the highest power of any of the unknowns is 1 (i.e. just x or y ). A quadratic equation is where the highest power of any of the unknowns is 2 (squared; i.e. x2 or y2 ). For example, solve the simultaneous equation below:

How do you find the root of a quadratic equation?

Open the quadratic equation solve mode: MODE 5 3 Enter the values of a, b and c. Use ? or = to move between the cells. Press = to solve the equation. The roots will be shown on screen. Use ? and ? to toggle between the roots.

Which axis corresponds to a quadratic equation ax2+bx+c=0?

The horizontal line, thexaxis, corresponds to points on the graph wherey= 0. So points wherethe graph touches or crosses this axis correspond to solutions of ax2+bx+c=0. In Figure 2, the graph in (a) never cuts or touches the horizontal axis and so this correspondsto a quadratic equationax2+bx+c= 0 having no real roots.

Chapter 1 Quadratic Equations

1.1 Equation:

values of the variable. For example

5x + 6 = 2, x is the variable (unknown)

The equations can be divided into the following two kinds:

Conditional Equation:

It is an equation in which two algebraic expressions are equal for particular value/s of the variable e.g., a) 2x = 3 is true only for x = 3/2 b) x

2 + x 6 = 0 is true only for x = 2, -3

Note: for simplicity a conditional equation is called an equation.

Identity:

It is an equation which holds good for all value of the variable e.g; a) (a + b) x ax + bx is an identity and its two sides are equal for all values of x. b) (x + 3) (x + 4) x

2 + 7x + 12 is also an identity which is true for all values of x.

1.2 Degree of an Equation:

The degree of an equation is the highest sum of powers of the variables in one of the term of the equation. For example

2x + 5 = 0 1

st degree equation in single variable

3x + 7y = 8 1

st degree equation in two variables 2x

2 7x + 8 = 0 2nd degree equation in single variable

2xy 7x + 3y = 2 2nd degree equation in two variables

x3 2x2 + 7x + 4 = 0 3rd degree equation in single variable x

2y + xy + x = 2 3rd degree equation in two variables

1.3 Polynomial Equation of Degree n:

An equation of the form

a nxn + an-1xn-1 + ---------------- + a3x3 + a2x2 + a1x + a0 = 0--------------(1)

Where n is a non-negative integer and a

n, an-1, -------------, a3, a2, a1, a0 are real constants, is called polynomial equation of degree n. Note that the degree of the equation in the single variable is the highest power of x which appear in the equation. Thus

3x4 + 2x3 + 7 = 0

4 + x3 + x2 + x + 1 = 0 , x4 = 0

are all fourth-degree polynomial equations. By the techniques of higher mathematics, it may be shown that nth degree equation of the form (1) has exactly n solutions (roots). These roots may be real, complex or a mixture of both. Further it may be shown that if such an equation has complex roots, they occur in pairs of conjugates complex numbers. In other words it cannot have an odd number of complex roots. A number of the roots may be equal. Thus all four roots of x4 = 0 are equal which are zero, and the four roots of x

4 2x2 + 1 = 0

Comprise two pairs of equal roots (1, 1, -1, -1).

Chapter 1 Quadratic Equations

1.4 Linear and Cubic Equation:

The equation of first degree is called linear equation.

For example,

i) x + 5 = 1 (in single variable) ii) x + y = 4 (in two variables) The equation of third degree is called cubic equation.

For example,

i) a3x3 + a2x2 + a1x + a0 = 0 (in single variable) ii) 9x3 + 5x2 + 3x = 0 (in single variable) iii) x2y + xy + y = 8 (in two variables)

1.5 Quadratic Equation:

The equation of second degree is called quadratic equation. The word quadratic ce the highest power of the unknown that appears in the equation is square. For example 2x

2 3x + 7 = 0 (in single variable)

xy 2x + y = 9 (in two variable)

Standard form of quadratic equation

The standard form of the quadratic equation is ax

2 + bx + c = 0, where a, b and c

are constants with a 0. If b 0 then this equation is called complete quadratic equation in x. If b = 0 then it is called a pure or incomplete quadratic equation in x.

For example, 5 x

2 + 6 x + 2 = 0 is a complete quadratic equation is x.

and 3x

2 4 = 0 is a pure or incomplete quadratic equation.

1.6 Roots of the Equation:

The value of the variable which satisfies the equation is called the root of the equation. A quadratic equation has two roots and hence there will be two values of the variable which satisfy the quadratic equation. For example the roots of x

2 + x 6 = 0

are 2 and -3.

1.7 Methods of Solving Quadratic Equation:

There are three methods for solving a quadratic equation: i) By factorization ii) By completing the square iii) By using quadratic formula i) Solution by Factorization:

Method:

Step I: Write the equation in standard form.

Step II: Factorize the quadratic equation on the left hand side if possible. Step III: The left hand side will be the product of two linear factors. Then equate each of the linear factor to zero and solve for values of x. These values of x give the solution of the equation.

Example 1

Solve the equation 3x

2 + 5x = 2

Solution:

2 + 5x = 2

Write in standard form 3x

2 + 5x 2 = 0

Chapter 1 Quadratic Equations

Factorize the left hand side 3x

2 + 6x x 2 = 0

3x(x + 2) -1(x + 2) = 0

(3x 1) (x + 2) = 0

Equate each of the linear factor to zero.

3x 1 = 0 or x + 2 = 0

3x = 1 or x = -2

x = 1 3 x = , - 2 are the roots of the Equation.

Solution Set = {, -2}

Example 2:

Solve the equation 6x

2 5x = 4

Solution:

2 5x = 4

2 5x 4 = 0

2 8x + 3x 4 = 0

2x (3x 4) +1 (3x 4) = 0

(2x + 1) (3x 4) = 0

Either 2x + 1 = 0 or 3x 4 = 0

Which gives 2x = 1 which gives 3x = 4

x = 1

2 x = 4

3 Required Solution Set =

14,23 ii) Solution of quadratic equation by Completing the

Square

Method:

Step I: Write the quadratic equation is standard form. Step II: Divide both sides of the equation by the co-efficient of x

2 if it is

not already 1.

Step III: Shift the constant term to the R.H.S.

Step IV: Add the square of one-half of the co-efficient of x to both side. Step V: Write the L.H.S as complete square and simplify the R.H.S. Step VI: Take the square root on both sides and solve for x.

Example 3:

Solve the equation 3x

2 = 15 4x by completing the square.

Solution: 3x2 = 15 4x

Step I Write in standard form: 3x

2 + 4x 15 = 0

Step II Dividing by 3 to both sides: x

2 + x 5 = 0

Chapter 1 Quadratic Equations

Step III Shift constant term to R.H.S: x

2 + 4

3x = 5

Step IV Adding the square of one half of the co-efficient of x. i.e., 24
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[PDF] Chapter 1 Quadratic Equations These values of x give

How to solve the quadratic equation with example?

What is the difference between linear and quadratic equations?

How do you find the root of a quadratic equation?

Which axis corresponds to a quadratic equation ax2+bx+c=0?

Chapter 1 Quadratic Equations

1.1 Equation:

5x + 6 = 2, x is the variable (unknown)

Conditional Equation:

2 + x 6 = 0 is true only for x = 2, -3

Identity:

2 + 7x + 12 is also an identity which is true for all values of x.

1.2 Degree of an Equation:

2x + 5 = 0 1

3x + 7y = 8 1

2 7x + 8 = 0 2nd degree equation in single variable

2xy 7x + 3y = 2 2nd degree equation in two variables

2y + xy + x = 2 3rd degree equation in two variables

1.3 Polynomial Equation of Degree n:

An equation of the form

Where n is a non-negative integer and a

3x4 + 2x3 + 7 = 0

4 + x3 + x2 + x + 1 = 0 , x4 = 0

4 2x2 + 1 = 0

Comprise two pairs of equal roots (1, 1, -1, -1).

1.4 Linear and Cubic Equation:

For example,

For example,

1.5 Quadratic Equation:

2 3x + 7 = 0 (in single variable)

Standard form of quadratic equation

The standard form of the quadratic equation is ax

2 + bx + c = 0, where a, b and c

For example, 5 x

2 + 6 x + 2 = 0 is a complete quadratic equation is x.

2 4 = 0 is a pure or incomplete quadratic equation.

1.6 Roots of the Equation:

2 + x 6 = 0

1.7 Methods of Solving Quadratic Equation:

Method:

Step I: Write the equation in standard form.

Example 1

Solve the equation 3x

2 + 5x = 2

Solution:

2 + 5x = 2

Write in standard form 3x

2 + 5x 2 = 0

Factorize the left hand side 3x

2 + 6x x 2 = 0

3x(x + 2) -1(x + 2) = 0

Equate each of the linear factor to zero.

3x 1 = 0 or x + 2 = 0

3x = 1 or x = -2

Solution Set = {, -2}

Example 2:

Solve the equation 6x

2 5x = 4

Solution:

2 5x = 4

2 5x 4 = 0

2 8x + 3x 4 = 0

2x (3x 4) +1 (3x 4) = 0

Either 2x + 1 = 0 or 3x 4 = 0

Which gives 2x = 1 which gives 3x = 4

2 x = 4

3 Required Solution Set =

Square

Method:

2 if it is

Step III: Shift the constant term to the R.H.S.

Example 3:

Solve the equation 3x

2 = 15 4x by completing the square.

Solution: 3x2 = 15 4x

Step I Write in standard form: 3x

2 + 4x 15 = 0

Step II Dividing by 3 to both sides: x

2 + x 5 = 0

Step III Shift constant term to R.H.S: x