Question

43. Find the roots of the quadratic equation 3x^2- 4x + 1 = 0 by the method of perfect square.​

Answer 1

Roots are (1 & ⅓)

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Answer 2

Answer:

The perfect square formula is used to find the square of the addition or subtraction of two terms (a ± b)$^{2}$ and is known as the perfect square formula.

Explanation:

The perfect square formula is used to find the square of the addition or subtraction of two terms (a ± b)$^{2}$ and is known as the perfect square formula.

Examples of the Perfect Square Formula

Let's look at some illustrations based on the formula $(a+b)^2$ in this part of the solved examples.

"Quad" means four, but "Quadratic" means "to make a square." A quadratic equation in its standard form is represented as:

$ax^2$ + bx + c = 0, where a, b and c are real numbers such that a ≠ 0, and x is a variable.

Since the degree of the equation written above is two, it will have two roots or solutions. The roots of polynomials are the values ​​of x that satisfy the equation. There are several methods for finding the roots of a quadratic equation. One of them is the completion of the square.

Example 1: Find the square of 6x + 4y using perfect formulas.

Solution:

To find the square of 6x + 4y,

Using the perfect square formula

$(a + b)^2 = (a^2 + 2ab + b^2)$

Enter the values,

$(6x + 4y)^2 = ((6x)^2 + 2*6x *4y + (4y)^2)$

$(6x + 4y)^2= (36x^2 + 48x + 16y^2)$

Answer: The square of 6x + 4y is $(36x^2 + 48x + 16y^2).$

Example 2: Use the perfect square formula to determine whether x2 + 25 - 10x is a perfect square or not.

Solution:

To find:$x^2 + 25 - 10x$ is a perfect square or not.

Rearrangement of Terms:

$x^2 + 25 -10x = x^2 + 5 *5 -2*5 *x = x^2 -2 *5*x + 5*5$

Using the perfect square formula.

$(a - b)^2 = (a^2 - 2ab + b^2)$

By comparing the values,

$x^2 - 2*5*x + 5*5 = (x - 5)^2$

According to the question:

$3x^2- 4x + 1 = 0\\3x^2-3x-x+1=0\\3x(x-1)-1(x-1)=0\\(3x-1)(x-1)=0\\x=\frac{1}{3} ,1$

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