Solve 1+4sqrt{1-x}=4x

$$1+4 \sqrt{ 1-x } =4x$$

Solve for x

$$x=\frac{3}{4}=0.75$$

Solution Steps

Subtract 1 from both sides of the equation.

$$4\sqrt{1-x}=4x-1$$

Square both sides of the equation.

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

Expand \left(4\sqrt{1-x}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{1-x}\right)^{2}=\left(4x-1\right)^{2}$$

Calculate \sqrt{1-x} to the power of 2 and get 1-x.

$$16\left(1-x\right)=\left(4x-1\right)^{2}$$

Use the distributive property to multiply 16 by 1-x.

$$16-16x=\left(4x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-1\right)^{2}.

$$16-16x=16x^{2}-8x+1$$

Subtract 16x^{2} from both sides.

$$16-16x-16x^{2}=-8x+1$$

Add 8x to both sides.

$$16-16x-16x^{2}+8x=1$$

Combine -16x and 8x to get -8x.

$$16-8x-16x^{2}=1$$

Subtract 1 from both sides.

$$16-8x-16x^{2}-1=0$$

Subtract 1 from 16 to get 15.

$$15-8x-16x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-16x^{2}-8x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -16x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-16\times 15=-240$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Calculate the sum for each pair.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

The solution is the pair that gives sum -8.

$$a=12 b=-20$$

Rewrite -16x^{2}-8x+15 as \left(-16x^{2}+12x\right)+\left(-20x+15\right).

$$\left(-16x^{2}+12x\right)+\left(-20x+15\right)$$

Factor out 4x in the first and 5 in the second group.

$$4x\left(-4x+3\right)+5\left(-4x+3\right)$$

Factor out common term -4x+3 by using distributive property.

$$\left(-4x+3\right)\left(4x+5\right)$$

To find equation solutions, solve -4x+3=0 and 4x+5=0.

$$x=\frac{3}{4} x=-\frac{5}{4}$$

Substitute \frac{3}{4} for x in the equation 1+4\sqrt{1-x}=4x.

$$1+4\sqrt{1-\frac{3}{4}}=4\times \frac{3}{4}$$

Simplify. The value x=\frac{3}{4} satisfies the equation.

$$3=3$$

Substitute -\frac{5}{4} for x in the equation 1+4\sqrt{1-x}=4x.

$$1+4\sqrt{1-\left(-\frac{5}{4}\right)}=4\left(-\frac{5}{4}\right)$$

Simplify. The value x=-\frac{5}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$7=-5$$

Equation 4\sqrt{1-x}=4x-1 has a unique solution.

$$x=\frac{3}{4}$$