Learning Task 3. Using the concept map below explain what you havelearned in thi...

Questions

September 2022 1 6 Report

Learning Task 3. Using the concept map below explain what you have
learned in this module.

1.) How can you solve
quadratic equation in one variable
using extracting the square roots?
How can you solve
quadratic equation in one variable
using factoring method?

2.)How do you illustrate
quadratic equation
in one variable?

3.)How can you solve
quadratic equation in one variable
using completing the square method?

4.)How can you solve
quadratic equation in one variable
using quadratic formula?

5.)How can you solve
quadratic equation in one variable using extracting the square roots?

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Answers & Comments

seraquesera

Verified answer

1. a. Extracting Square Roots

We could instead isolate the perfect square x2, then take the square root of both sides of the equation to solve for x. This alternative method does not need factoring. Extracting square roots is the term used for this.

Finding the Square Roots of Quadratic Equations:

A quadratic equation of the type ax2 + c = 0 can be resolved by identifying the perfect square that contains the variable x and finding the square roots of both sides.

$ax^{2} + c = 0$

$ax^{2} = -c$

$x^{2} = \frac{-c}{a}$

$x =$ ± $\sqrt{\frac{-c}{a} }$

Keep in mind that you can use square roots to solve any of the following quadratic equations:

$(5x)^{2} - 20 = 0$

$(x + 1)^{2} - 9 = 0$

$3 (x - 4)^{2} - 16 = 0$

This approach can be used to resolve quadratic equations as long as we can identify the perfect square containing the variable x and calculate the square roots of both sides of the equation.

Extracting square roots yields the same result as factoring, as seen on the preceding page.

$49 - (2x)^{2} = 0$

$49 = (2x)^{2}$

± $\sqrt{49} = \sqrt{(2x)^{2} }$

± $7 = 2x$

± $\frac{7}{2} = x$

Same as,

$49 - (2x)^{2} = 0$

$(7 - 2x)(7 +2x) = 0$

$7 - 2x = 0 ; 7 + 2x = 0$

$7 = 2x ; 7 = -2x$

$\frac{7}{2} = x ; -\frac{7}{2} = x$

b. Factoring

1. ALWAYS check for a GCF first when factoring. No matter how you factor or what kind of polynomial you have, this needs to be done (binomial, trinomial ...)

2. If the polynomial (binomial) has two terms, determine whether they are both perfect squares or perfect cubes.

a. Use the difference of squares formula if the two terms are perfect squares and are being subtracted.

i. $x^{2} - y^{2} = (x + y)(x - y)$

b. When adding or subtracting two words that are perfect cubes, use the sum or difference of cubes formulae.

i. $x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2} )$

ii. $x^{3} + y^{3} = (x + y)(x^{2} - xy + y^{2} )$

c. A binomial is not factorable if none of the aforementioned conditions are met.

3. Apply the ac-method if the polynomial has three terms (a trinomial).

4. Factor by grouping if the polynomial has four terms.

No matter how you factor, ALWAYS verify that your factors can be factored in and ALWAYS factor fully.

2. Construction of a quadratic equation in a single variable

The roots of the equation are the answers to quadratic or polynomial equations. It is known as the construction of a quadratic equation in one variable when there is only one variable, x, in the equation. Equations that are quadratic differ from equations that are linear. We can solve quadratic equations using a variety of techniques, including factoring, squaring, and the quadratic formula.

f(x) has a quadratic expression.

Two separate real zeros either there could be two equal real zeros.

It could not have genuine zeros

f(x) = 0 is a quadratic equation that

Two separate real zeros, either
No real zeros and two equal real zeros

3. Completing the square

The constant term c is ALWAYS equal to half the leading coefficient of x squared when a perfect square trinomial is expressed in polynomial form and the leading coefficient is 1.

First, we must confirm that the quadratic equation we are dealing with has a leading coefficient of 1. To obtain the constant term required to have a perfect square trinomial, we must next take the coefficient of x, divide it in half, and then square it.

How to Finish the Square in Steps:

Multiply each term by the leading coefficient (a); if a = 1, omit this step.
Clearly define the constant term (c)
Increase both sides $(\frac{b}{2}) ^{2}$ By the square of the inverse x coefficient.
Use a perfect square trinomial to factor the polynomial.
Get a solution by removing square roots.

4. Quadratic Formula

Ax2 + bx + c = 0 is the quadratic equation in standard form. Determine what a, b, and c are worth.
The quadratic formula in writing. After that, enter the values for a, b, and c.
Simplify.
Verify the answers.

You can quickly memorize the formula if you speak it out loud as you write it in each problem. The Quadratic Formula is an EQUATION, therefore keep that in mind. Ensure that you begin with "x =".