Question

How do you solve `sqrt(3 - x)= 2x`?

Answer 1

`x=-1, 3/4`

Explanation:

`sqrt(3-x)=2x`

First square both sides.

`(sqrt(3-x))^2=(2x)^2`

`3-x=4x^2`

Move all terms to the left side.

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

Rewrite the expression in standard form.

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

This is a quadratic equation in standard form , `ax+bx+c`, where `a=-4, b=-1, c=3`.

Use the a·c method to solve.

Multiply `axxc`.

`-4xx3=-12`

Find two numbers that when multiplied equal `-12`, and when added equal `-1`.

`-4 and 3` fit the pattern.

Rewrite the equation with `-4x and 3x` in place of `-x`.

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

Factor out common terms in the first pair of terms and in the second pair of terms.

`-4x(x+1)+3(x+1)=0`

Factor out `(x+1)`.

`color(red)((x+1)color(blue)((-4x+3)color(black)(=0)`

Set each term in parentheses equal to zero and solve for `x`.

`color(red)(x+1=0)`

`color(red)(x=-1)`

`color(blue)(-4x+3=0)`

`color(blue)(-4x=-3)`

Divide both sides by `color(blue)(-4)`.

`color(blue)(x=3/4)`

Answer 2

`x=-1, 3/4`

Explanation:

`sqrt(3-x)=2x`?

Square both sides

`(sqrt(3-x))^2=(2x)^2`

Simplify.

`3-x=4x^2`

Move all terms to the left side.

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

Rewrite in standard form.

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

This is a quadratic equation in standard form, `ax+bx+3`, where `a=-4, b=-1, c=3`

We can find the values for `x` using the quadratic formula.

Quadratic Formula

`x=(-b+-sqrt(b^2-4ac))/(2a)`

Substitute the known values into the formula.

`x=(-(-1)+-sqrt((-1)^2-(4*-4*3)))/(2*-4)`

Simplify.

`x=1+-sqrt(1-(-48))/(-8)`

Simplify.

`x=(1+-sqrt(49))/(-8)`

Simplify.

`color(purple)(x=(1+-7)/(-8))`

`color(red)(x=(1+7)/(-8))`

`color(red)(x=8/(-8)`

`color(red)(x=-1)`

`color(blue)(x=(1-7)/(-8))`

`color(blue)(x=(-6)/(-8))`

`color(blue)(x=6/8)`

Simplify.

`color(blue)(x=3/4)`