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

Jan 31, 2016

$x = - 1 , \frac{3}{4}$

#### Explanation:

$\sqrt{3 - x} = 2 x$

First square both sides.

${\left(\sqrt{3 - x}\right)}^{2} = {\left(2 x\right)}^{2}$

$3 - x = 4 {x}^{2}$

Move all terms to the left side.

$3 - x - 4 {x}^{2} = 0$

Rewrite the expression in standard form.

$- 4 {x}^{2} - x + 3 = 0$

This is a quadratic equation in standard form , $a x + b x + c$, where $a = - 4 , b = - 1 , c = 3$.

Use the a·c method to solve.

Multiply $a \times c$.

$- 4 \times 3 = - 12$

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

$- 4 \mathmr{and} 3$ fit the pattern.

Rewrite the equation with $- 4 x \mathmr{and} 3 x$ in place of $- x$.

$- 4 {x}^{2} - 4 x + 3 x + 3 = 0$

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

$- 4 x \left(x + 1\right) + 3 \left(x + 1\right) = 0$

Factor out $\left(x + 1\right)$.

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

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

$\textcolor{red}{x + 1 = 0}$

$\textcolor{red}{x = - 1}$

$\textcolor{b l u e}{- 4 x + 3 = 0}$

$\textcolor{b l u e}{- 4 x = - 3}$

Divide both sides by $\textcolor{b l u e}{- 4}$.

$\textcolor{b l u e}{x = \frac{3}{4}}$

Jan 31, 2016

$x = - 1 , \frac{3}{4}$

#### Explanation:

$\sqrt{3 - x} = 2 x$?

Square both sides

${\left(\sqrt{3 - x}\right)}^{2} = {\left(2 x\right)}^{2}$

Simplify.

$3 - x = 4 {x}^{2}$

Move all terms to the left side.

$- 4 {x}^{2} + 3 - x = 0$

Rewrite in standard form.

$- 4 {x}^{2} - x + 3 = 0$

This is a quadratic equation in standard form, $a x + b x + 3$, where $a = - 4 , b = - 1 , c = 3$

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

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Substitute the known values into the formula.

$x = \frac{- \left(- 1\right) \pm \sqrt{{\left(- 1\right)}^{2} - \left(4 \cdot - 4 \cdot 3\right)}}{2 \cdot - 4}$

Simplify.

$x = 1 \pm \frac{\sqrt{1 - \left(- 48\right)}}{- 8}$

Simplify.

$x = \frac{1 \pm \sqrt{49}}{- 8}$

Simplify.

$\textcolor{p u r p \le}{x = \frac{1 \pm 7}{- 8}}$

$\textcolor{red}{x = \frac{1 + 7}{- 8}}$

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

$\textcolor{red}{x = - 1}$

$\textcolor{b l u e}{x = \frac{1 - 7}{- 8}}$

$\textcolor{b l u e}{x = \frac{- 6}{- 8}}$

$\textcolor{b l u e}{x = \frac{6}{8}}$

Simplify.

$\textcolor{b l u e}{x = \frac{3}{4}}$