Chapter 9.5, Problem 7CYU

To determine

Find the solution.

Answer to Problem 7CYU

$x=-2.4$ or $x=0.1$

Explanation of Solution

Given Information: $4x^2+9x-1=0$

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

Calculation:

  $4x^2+9x-1=0$

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

  $x^2+\frac{9}{4}x-\frac{1}{4}=0$

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

  $x^2+\frac{9}{4}x-\frac{1}{4}+{\left(\frac{9}{8}\right)}^2-{\left(\frac{9}{8}\right)}^2=0$

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

  ${\left(x+\frac{9}{8}\right)}^2-\frac{1}{4}-{\left(\frac{9}{8}\right)}^2=0$

So,

  ${\left(x+\frac{9}{8}\right)}^2-\frac{1}{4}-\frac{81}{64}=0$

So,

  ${\left(x+\frac{9}{8}\right)}^2-\left(\frac{16+81}{64}\right)=0$

Hence,

  ${\left(x+\frac{9}{8}\right)}^2-\left(\frac{97}{64}\right)=0$

Now, square root of 97 is $\sqrt{97}$ and square root of 64 is 8.

  ${\left(x+\frac{9}{8}\right)}^2-{\left(\frac{\sqrt{97}}{8}\right)}^2=0$

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

  $\left(x+\frac{9}{8}+\frac{\sqrt{97}}{8}\right)\left(x+\frac{9}{8}-\frac{\sqrt{97}}{8}\right)=0$

Now,

$\left(x+\frac{9}{8}+\frac{\sqrt{97}}{8}\right)=0$ or $\left(x+\frac{9}{8}-\frac{\sqrt{97}}{8}\right)=0$

Now,

$x+\frac{9+\sqrt{97}}{8}=0$ or $x+\frac{9-\sqrt{97}}{8}=0$

Now,

$x+2.35=0$ or $x-0.11=0$

Hence,

$x=-2.35$ or $x=0.11$

Nearest Tenth $x=-2.4$ or $x=0.1$