Solve sqrt{5x^2-6x-4}=2(x-1)

$$\sqrt { 5 x ^ { 2 } - 6 x - 4 } = 2 ( x - 1 )$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5x^{2}-6x-4}\right)^{2}=\left(2\left(x-1\right)\right)^{2}$$

Calculate \sqrt{5x^{2}-6x-4} to the power of 2 and get 5x^{2}-6x-4.

$$5x^{2}-6x-4=\left(2\left(x-1\right)\right)^{2}$$

Use the distributive property to multiply 2 by x-1.

$$5x^{2}-6x-4=\left(2x-2\right)^{2}$$

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

$$5x^{2}-6x-4=4x^{2}-8x+4$$

Subtract 4x^{2} from both sides.

$$5x^{2}-6x-4-4x^{2}=-8x+4$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}-6x-4=-8x+4$$

Add 8x to both sides.

$$x^{2}-6x-4+8x=4$$

Combine -6x and 8x to get 2x.

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

Subtract 4 from both sides.

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

Subtract 4 from -4 to get -8.

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

To solve the equation, factor x^{2}+2x-8 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=2 ab=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 2.

$$a=-2 b=4$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-2\right)\left(x+4\right)$$

To find equation solutions, solve x-2=0 and x+4=0.

$$x=2 x=-4$$

Substitute 2 for x in the equation \sqrt{5x^{2}-6x-4}=2\left(x-1\right).

$$\sqrt{5\times 2^{2}-6\times 2-4}=2\left(2-1\right)$$

Simplify. The value x=2 satisfies the equation.

$$2=2$$

Substitute -4 for x in the equation \sqrt{5x^{2}-6x-4}=2\left(x-1\right).

$$\sqrt{5\left(-4\right)^{2}-6\left(-4\right)-4}=2\left(-4-1\right)$$

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

$$10=-10$$

Equation \sqrt{5x^{2}-6x-4}=2\left(x-1\right) has a unique solution.

$$x=2$$