Question

How do you solve `4x^2+3=8x`?

Answer 1

The solutions to the equation are `x=1/2` and `x=3/2`.

Explanation:

We know that this is a quadratic since it has an `x^2` term in it.

To solve quadratics, first set one side equal to `0`:

`4x^2+3=8x`

`4x^2+3-8x=0`

`4x^2-8x+3=0`

Now, we have to find two numbers that multiply to `12` (the `a` term times the `c` term) and add up to `-8` (the `b` term).

After some experimentation, you will find that these two numbers are `-6` and `-2`.

Now, split up the `x` terms into these two numbers, then factor the first two terms and the last two terms separately, and lastly, combine them:

`4x^2-6x-2x+3=0`

`color(red)(2x)(2x-3)-2x+3=0`

`color(red)(2x)(2x-3)-color(blue)1(2x-3)=0`

`(color(red)(2x)-color(blue)1)(2x-3)=0`

Now, solve for when each factor equals `0`, and those will be the solutions:

#color(white){color(black)( (2x-1=0,qquadqquad2x-3=0), (2x=1,qquadqquad2x=3), (x=1/2,qquadqquadx=3/2):}#

Those are the answers. We can check out answers by graphing the quadratic `4x^2-8x+3` on a calculator and seeing where it crosses the `x`-axis (it should cross when `x=1/2` and also when `x=3/2`):

graph{4x^2-8x+3 [-0.5, 2.5, -1.29, 0.419]}

It does, so our answers are correct. Hope this helped!

Answer 2

`x=1/2` and `x=3/2`

Explanation:

Solve the equation by the new Transforming Method (Socratic Search).

`y = 4x^2 - 8x + 3 = 0`

Transformed equation:

`y' = x^2 - 8x + 12`.

Proceeding. Find `2` real roots of `y'`, then, divide them by `a = 4`.

Find `2` real roots knowing the sum `(-b = 8)` and the product
`(ac = 12)`. They are: `2` and `6`.

The `2` real roots of `y` are:

`x_1 = 2/a = 2/4 = color(blue)(1/2)` and `x_2 = 6/a = 6/4 = color(blue)(3/2)`.