Math

Roots

Roots

• The value/s which make/s the quadratic equation true.

Discriminant

• This will guide us in determining the nature of the roots without solving.

• b² - 4ac

Finding the roots by Quadratic Formula

• if b² - 4ac > 0, the equation has two different real roots.

Finding the roots by Quadratic Formula

• if b² - 4ac = 0, the equation has one real root.

Finding the roots by Quadratic Formula

• if b² - 4ac < 0, the equation has no real roots.

Quadratic Term

• The term with degree of 2 in a quadratic equation.

• ± ax²

Linear Term

• The term with the degree of 1 in a quadratic equation.

• ± bx

Factoring

• A method of solving quadratic equations which uses common monomial factor, difference of two squares, perfect square trinomials and general quadratic trinomials.

Solutions

• The other term given to roots of quadratic equation.

Zero Product Property

• The property which states that if a and b are real numbers such that ab = 0, then a = 0 and b = 0.

Square Root Property

• The property which states that, for any non-negative real number c, if x² = c, then x = ± √c.

Extraneous

• A value which was solved but is not a solution to the quadratic equation.

Imaginary

• A root of a quadratic equation which is not real.

Constant Term

• The term with a degree of zero in a quadratic equation.

• ± c

Rational root

• The nature of root of a quadratic equation wherein the discriminant is a perfect square.

Irrational root

• The nature of root of a quadratic equation wherein the discriminant is not a perfect square.

Zero nature of roots

• The value solved for the discriminant of a quadratic equation which has real and equal roots.

Positive nature of roots

• The value solved for the discriminant of a quadratic equation which has real and unequal roots.

Negative nature of roots

• The value solved for the discriminant of a quadratic equation which has not real roots.

Standard/General Form

• The form in which a quadratic equation is written as ax² + bx + c = 0.

Quadratic

• Comes from the Latin word quadratus which means “to make a square”.

Steps in Solving Quadratic Equations by Factoring

1) Write the equation in standard form, with the right member zero.

2) Factor the left member of the equation.

3) Set each factor equal to zero.

4) Solve the resulting first-degree equations.

5) Check if the answers are correct by substituting them to the variable in the original equation.

Steps in Finding the Roots by Completing the Square.

1) If the value of a is 1, proceed to step 2. Otherwise, divide both sides of the equation by a.

2) Group all variable terms on one side of the equation and constant on the other side, x² + bx = c.

3) Complete the square of the resulting binomial by adding on both sides of the equation the square of half of b. x² + bx + (1/2 b)² = c + (1/2 b)².

4) Factor the resulting perfect square trinomial and write it as square of binomial. (x + b/2)².

5) Use the Square Root Property to solve for x. (x + b/2) = √c + (c + (b/2)².

Sum and Product of the Roots of a Quadratic Equation



If r₁ and r₂ are the roots of the quadratic equation ax² + bx + c = 0, then:

• Sum of the root: r₁ + r₂ = -b/a.

• Products of the roots: r₁ ⋅ r₂ = c/a.

If we divide ax² + bx + c = 0 by a, we obtain x² + bx/a + c/a = 0.

The sum of the roots of ax² + bx + c = 0 is -b/a and the product of roots is c/a.



Sum and Product of the Roots of a Quadratic Equation

A quadratic equation can be written in the form:

x² - (sum of the roots)x + (product of the roots) = 0.



Steps in finding the two consecutive positive even integers.

1) Identify the conditions stated.

2) Represent the unknown.

3) Form the equation satisfying the conditions mentioned in the problem.

4) Solve the resulting equation using the appropriate method.

5) Identify extraneous solutions if there are.

6) State the answer.

In finding two consecutive numbers with the given product, think of the factors of the product which have a difference of 1.

In finding the two consecutive even (or odd) numbers with the given product, think of the even (or odd) factors of the product which have a difference of 2.

Finding the consecutive numbers.

How to find the sum of two consecutive numbers. (refer to the picture)

Finding the consecutive numbers.

How to find the product of two consecutive numbers, (refer to the picture)

Finding the dimensions of a rectangle.

How to find the dimensions of a rectangle. (refer to the picture)

Area = length x width