Question

Using the completing the square method solve for 4x squared minus 8x minus 5 equal to 04x² - 8x - 5 = 0

Answer 1

ANSWER

x = -1/2 and x = 5/2

EXPLANATION

The method of completing the square consists of rewriting a quadratic equation as a binomial squared,

$(a\pm b)^2=a^2\pm2ab+b^2$

First, we have to identify the first term, a. Note that in the given equation, we have 4x², and 4 is 2², so this term is equivalent to,

$4x^2=(2x)^2$

Therefore, our first term is 2x.

Then, we have to find the second term, b. In the given equation, the second term is -8x, which is equal to 2ab in the binomial squared rule. We know that a = 2x, so now we can find b,

$\begin{gathered} 2ab=-8x \\ \\ 2\cdot2x\cdot b=-8x \end{gathered}$

Solving for b,

$\begin{gathered} 4xb=-8x \\ \\ b=\frac{-8x}{4x}=-2 \end{gathered}$

Now, write the binomial and expand it with the rule stated at the top of the Explanation section,

$(2x-2)^2=4x^2-8x+4$

The result is not equal to the given equation, so, to be able to replace the first two terms of the equation for this binomial squared, we have to add 4 to both sides of the equation,

$\begin{gathered} 4x^2-8x+4-5=0+4 \\ \\ (2x-2)^2-5=4 \end{gathered}$

Finally, we have to solve for x. First, add 5 to both sides,

$\begin{gathered} (2x-2)^2-5+5=4+5 \\ \\ (2x-2)^2=9 \end{gathered}$

Then, take the square root of both sides,

$\begin{gathered} \sqrt{(2x-2)^2}=\pm\sqrt{9} \\ \\ 2x-2=\pm3 \end{gathered}$

Add 2 to both sides,

$\begin{gathered} 2x-2+2=2\pm3 \\ \\ 2x=2\pm3 \end{gathered}$

And then, divide both sides by 2,

$x=\frac{2\pm3}{2}$

So, we have the two solutions,

$\begin{gathered} x_1=\frac{2+3}{2}=\frac{5}{2} \\ \\ x_2=\frac{2-3}{2}=-\frac{1}{2} \end{gathered}$

Hence, the two solutions are x = -1/2 and x = 5/2.