Solve sqrt{7x^2-5x+2}=3x-1

$$\sqrt { 7 x ^ { 2 } - 5 x + 2 } = 3 x - 1$$

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{7x^{2}-5x+2}\right)^{2}=\left(3x-1\right)^{2}$$

Calculate \sqrt{7x^{2}-5x+2} to the power of 2 and get 7x^{2}-5x+2.

$$7x^{2}-5x+2=\left(3x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$7x^{2}-5x+2=9x^{2}-6x+1$$

Subtract 9x^{2} from both sides.

$$7x^{2}-5x+2-9x^{2}=-6x+1$$

Combine 7x^{2} and -9x^{2} to get -2x^{2}.

$$-2x^{2}-5x+2=-6x+1$$

Add 6x to both sides.

$$-2x^{2}-5x+2+6x=1$$

Combine -5x and 6x to get x.

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

Subtract 1 from both sides.

$$-2x^{2}+x+2-1=0$$

Subtract 1 from 2 to get 1.

$$-2x^{2}+x+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=1 ab=-2=-2$$

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. The only such pair is the system solution.

$$a=2 b=-1$$

Rewrite -2x^{2}+x+1 as \left(-2x^{2}+2x\right)+\left(-x+1\right).

$$\left(-2x^{2}+2x\right)+\left(-x+1\right)$$

Factor out 2x in -2x^{2}+2x.

$$2x\left(-x+1\right)-x+1$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(2x+1\right)$$

To find equation solutions, solve -x+1=0 and 2x+1=0.

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

Substitute 1 for x in the equation \sqrt{7x^{2}-5x+2}=3x-1.

$$\sqrt{7\times 1^{2}-5+2}=3\times 1-1$$

Simplify. The value x=1 satisfies the equation.

$$2=2$$

Substitute -\frac{1}{2} for x in the equation \sqrt{7x^{2}-5x+2}=3x-1.

$$\sqrt{7\left(-\frac{1}{2}\right)^{2}-5\left(-\frac{1}{2}\right)+2}=3\left(-\frac{1}{2}\right)-1$$

Simplify. The value x=-\frac{1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{5}{2}=-\frac{5}{2}$$

Equation \sqrt{7x^{2}-5x+2}=3x-1 has a unique solution.

$$x=1$$