Solve sqrt{4x^2-6x+2}+sqrt{4x^2-1}=2sqrt{4x^2-4x+1}

$$\sqrt { 4 x ^ { 2 } - 6 x + 2 } + \sqrt { 4 x ^ { 2 } - 1 } = 2 \sqrt { 4 x ^ { 2 } - 4 x + 1 }$$

Solve for x

$$x = \frac{17}{16} = 1\frac{1}{16} = 1.0625$$

$$x=\frac{1}{2}=0.5$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}\right)^{2}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}\right)^{2}.

$$\left(\sqrt{4x^{2}-6x+2}\right)^{2}+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}+\left(\sqrt{4x^{2}-1}\right)^{2}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

Calculate \sqrt{4x^{2}-6x+2} to the power of 2 and get 4x^{2}-6x+2.

$$4x^{2}-6x+2+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}+\left(\sqrt{4x^{2}-1}\right)^{2}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

Calculate \sqrt{4x^{2}-1} to the power of 2 and get 4x^{2}-1.

$$4x^{2}-6x+2+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}+4x^{2}-1=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-6x+2+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}-1=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

Subtract 1 from 2 to get 1.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

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

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=2^{2}\left(\sqrt{4x^{2}-4x+1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=4\left(\sqrt{4x^{2}-4x+1}\right)^{2}$$

Calculate \sqrt{4x^{2}-4x+1} to the power of 2 and get 4x^{2}-4x+1.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=4\left(4x^{2}-4x+1\right)$$

Use the distributive property to multiply 4 by 4x^{2}-4x+1.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=16x^{2}-16x+4$$

Subtract 8x^{2}-6x+1 from both sides of the equation.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=16x^{2}-16x+4-\left(8x^{2}-6x+1\right)$$

To find the opposite of 8x^{2}-6x+1, find the opposite of each term.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=16x^{2}-16x+4-8x^{2}+6x-1$$

Combine 16x^{2} and -8x^{2} to get 8x^{2}.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=8x^{2}-16x+4+6x-1$$

Combine -16x and 6x to get -10x.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=8x^{2}-10x+4-1$$

Subtract 1 from 4 to get 3.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=8x^{2}-10x+3$$

Square both sides of the equation.

$$\left(2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

Expand \left(2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}\right)^{2}.

$$2^{2}\left(\sqrt{4x^{2}-6x+2}\right)^{2}\left(\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{4x^{2}-6x+2}\right)^{2}\left(\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

Calculate \sqrt{4x^{2}-6x+2} to the power of 2 and get 4x^{2}-6x+2.

$$4\left(4x^{2}-6x+2\right)\left(\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

Calculate \sqrt{4x^{2}-1} to the power of 2 and get 4x^{2}-1.

$$4\left(4x^{2}-6x+2\right)\left(4x^{2}-1\right)=\left(8x^{2}-10x+3\right)^{2}$$

Use the distributive property to multiply 4 by 4x^{2}-6x+2.

$$\left(16x^{2}-24x+8\right)\left(4x^{2}-1\right)=\left(8x^{2}-10x+3\right)^{2}$$

Use the distributive property to multiply 16x^{2}-24x+8 by 4x^{2}-1 and combine like terms.

$$64x^{4}+16x^{2}-96x^{3}+24x-8=\left(8x^{2}-10x+3\right)^{2}$$

Square 8x^{2}-10x+3.

$$64x^{4}+16x^{2}-96x^{3}+24x-8=64x^{4}-160x^{3}+148x^{2}-60x+9$$

Subtract 64x^{4} from both sides.

$$64x^{4}+16x^{2}-96x^{3}+24x-8-64x^{4}=-160x^{3}+148x^{2}-60x+9$$

Combine 64x^{4} and -64x^{4} to get 0.

$$16x^{2}-96x^{3}+24x-8=-160x^{3}+148x^{2}-60x+9$$

Add 160x^{3} to both sides.

$$16x^{2}-96x^{3}+24x-8+160x^{3}=148x^{2}-60x+9$$

Combine -96x^{3} and 160x^{3} to get 64x^{3}.

$$16x^{2}+64x^{3}+24x-8=148x^{2}-60x+9$$

Subtract 148x^{2} from both sides.

$$16x^{2}+64x^{3}+24x-8-148x^{2}=-60x+9$$

Combine 16x^{2} and -148x^{2} to get -132x^{2}.

$$-132x^{2}+64x^{3}+24x-8=-60x+9$$

Add 60x to both sides.

$$-132x^{2}+64x^{3}+24x-8+60x=9$$

Combine 24x and 60x to get 84x.

$$-132x^{2}+64x^{3}+84x-8=9$$

Subtract 9 from both sides.

$$-132x^{2}+64x^{3}+84x-8-9=0$$

Subtract 9 from -8 to get -17.

$$-132x^{2}+64x^{3}+84x-17=0$$

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

$$64x^{3}-132x^{2}+84x-17=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -17 and q divides the leading coefficient 64. List all candidates \frac{p}{q}.

$$±\frac{17}{64},±\frac{17}{32},±\frac{17}{16},±\frac{17}{8},±\frac{17}{4},±\frac{17}{2},±17,±\frac{1}{64},±\frac{1}{32},±\frac{1}{16},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=\frac{1}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 64x^{3}-132x^{2}+84x-17 by 2\left(x-\frac{1}{2}\right)=2x-1 to get 32x^{2}-50x+17. Solve the equation where the result equals to 0.

$$32x^{2}-50x+17=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 32 for a, -50 for b, and 17 for c in the quadratic formula.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 32\times 17}}{2\times 32}$$

Do the calculations.

$$x=\frac{50±18}{64}$$

Solve the equation 32x^{2}-50x+17=0 when ± is plus and when ± is minus.

$$x=\frac{1}{2} x=\frac{17}{16}$$

List all found solutions.

$$x=\frac{1}{2} x=\frac{17}{16}$$

Substitute \frac{1}{2} for x in the equation \sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}=2\sqrt{4x^{2}-4x+1}.

$$\sqrt{4\times \left(\frac{1}{2}\right)^{2}-6\times \frac{1}{2}+2}+\sqrt{4\times \left(\frac{1}{2}\right)^{2}-1}=2\sqrt{4\times \left(\frac{1}{2}\right)^{2}-4\times \frac{1}{2}+1}$$

Simplify. The value x=\frac{1}{2} satisfies the equation.

$$0=0$$

Substitute \frac{17}{16} for x in the equation \sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}=2\sqrt{4x^{2}-4x+1}.

$$\sqrt{4\times \left(\frac{17}{16}\right)^{2}-6\times \frac{17}{16}+2}+\sqrt{4\times \left(\frac{17}{16}\right)^{2}-1}=2\sqrt{4\times \left(\frac{17}{16}\right)^{2}-4\times \frac{17}{16}+1}$$

Simplify. The value x=\frac{17}{16} satisfies the equation.

$$\frac{9}{4}=\frac{9}{4}$$

List all solutions of \sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}=2\sqrt{4x^{2}-4x+1}.

$$x=\frac{1}{2} x=\frac{17}{16}$$