Chapter 9.6, Problem 2AGP

To determine

Find the solution.

Answer to Problem 2AGP

  $x=2.5$ or $x=3.5$

Explanation of Solution

Given Information: $4x^2-24x+35=0$

Formula Used: $a^2-2ab+b^2={(a-b)}^2$ and $a^2-b^2=(a+b)(a-b)$

Calculation:

  $4x^2-24x+35=0$

Now, to make coefficient of $x^2$ = 1, divide the equation by 4.

  $x^2-\frac{24}{4}x+\frac{35}{4}=0$

Now, adding & subtracting the square of half the coefficient of x.

  $x^2-6x+\frac{35}{4}+{\left(\frac{6}{2}\right)}^2-{\left(\frac{6}{2}\right)}^2=0$

Now use formula, $a^2-2ab+b^2={(a-b)}^2$

  ${\left(x-3\right)}^2+\frac{35}{4}-{\left(3\right)}^2=0$

So,

  ${\left(x-3\right)}^2+\frac{35}{4}-9=0$

Now,

  ${\left(x-3\right)}^2+\frac{35-36}{4}=0$

Now,

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

Now, square root of 1 is 1 and square root of 4 is 2

  ${\left(x-3\right)}^2-{\left(\frac{1}{2}\right)}^2=0$

Now use formula, $a^2-b^2=(a+b)(a-b)$

  $\left(x-3+\frac{1}{2}\right)\left(x-3-\frac{1}{2}\right)=0$

Now,

  $x=3-\frac{1}{2}$ or $x=3+\frac{1}{2}$

Now,

  $x=\frac{5}{2}$ or $x=\frac{7}{2}$

Hence,

  $x=2.5$ or $x=3.5$