Quadratic equation - What is the value of x?

Question

"Find out the value of x by this equation: $(x+a)^2 = (2a-3x)^2$". (The answer should by the way, according to my book, be

$x1 = 0.25a$

$x2 = 1.5a$

Here's how far I've gotten:

$(x+a)^2 = (2a-3x)^2$

$x^2 + 2ax + a^2 = 4a^2 - 12ax + 9x^2$

$x^2 = 3a^2 + 9x^2 - 14ax$

After that I'm stuck. I haven't done any problem like this (ever) before, so I don't know what to do. I did try to "formulate" the equation differently to see if I would have an easier time solving it like this:

$-8x^2 + 14ax - 3a^2 = 0$

But I honestly don't know what do after that. I'd love it if someone could thoroughly explain the steps I need to do in order to get the value of $x$.

Answer 1

No quadratic equation really required here:

\begin{align*} (x + a)^2 = (2a - 3x)^2 & \iff x + a = \pm (2a - 3x)\\ & \iff x + a = \begin{cases} 2a - 3x\\ 3x - 2a \end{cases}\\ & \iff x = \begin{cases} \dfrac{a}{4}\\[0.2 mm] \dfrac{3a}{2} \end{cases} \end{align*}

Answer 2

$8x^2-14ax+3a^2=0$

Use the quadratic equation

$x=\frac{14a±\sqrt{(14a)^2-4(8)(3a^2)}}{16}$

$=\frac{14a±10a}{16}$

etc.

Answer 3

Treat symbol $a$ as if it is a constant.

$$ -8x^2 + 14ax - 3a^2 = ( 4 x -a)( -2 x + 3 a)= 0, $$

$$ x = ( 3 a/2, a/4 ) $$