Solve 7x^2-x=2x-x^2

$$7 x ^ { 2 } - x = 2 x - x ^ { 2 }$$

Solve for x

$$x=\frac{3}{8}=0.375$$

$$x=0$$

Steps Using Factoring

Subtract 2x from both sides.

$$7x^{2}-x-2x=-x^{2}$$

Combine -x and -2x to get -3x.

$$7x^{2}-3x=-x^{2}$$

Add x^{2} to both sides.

$$7x^{2}-3x+x^{2}=0$$

Combine 7x^{2} and x^{2} to get 8x^{2}.

$$8x^{2}-3x=0$$

Factor out x.

$$x\left(8x-3\right)=0$$

To find equation solutions, solve x=0 and 8x-3=0.

$$x=0 x=\frac{3}{8}$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$7x^{2}-x-2x=-x^{2}$$

Combine -x and -2x to get -3x.

$$7x^{2}-3x=-x^{2}$$

Add x^{2} to both sides.

$$7x^{2}-3x+x^{2}=0$$

Combine 7x^{2} and x^{2} to get 8x^{2}.

$$8x^{2}-3x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}}}{2\times 8}$$

Take the square root of \left(-3\right)^{2}.

$$x=\frac{-\left(-3\right)±3}{2\times 8}$$

The opposite of -3 is 3.

$$x=\frac{3±3}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{3±3}{16}$$

Now solve the equation x=\frac{3±3}{16} when ± is plus. Add 3 to 3.

$$x=\frac{6}{16}$$

Reduce the fraction \frac{6}{16} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{8}$$

Now solve the equation x=\frac{3±3}{16} when ± is minus. Subtract 3 from 3.

$$x=\frac{0}{16}$$

Divide 0 by 16.

$$x=0$$

The equation is now solved.

$$x=\frac{3}{8} x=0$$

Steps for Completing the Square

Subtract 2x from both sides.

$$7x^{2}-x-2x=-x^{2}$$

Combine -x and -2x to get -3x.

$$7x^{2}-3x=-x^{2}$$

Add x^{2} to both sides.

$$7x^{2}-3x+x^{2}=0$$

Combine 7x^{2} and x^{2} to get 8x^{2}.

$$8x^{2}-3x=0$$

Divide both sides by 8.

$$\frac{8x^{2}-3x}{8}=\frac{0}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}-\frac{3}{8}x=\frac{0}{8}$$

Divide 0 by 8.

$$x^{2}-\frac{3}{8}x=0$$

Divide -\frac{3}{8}, the coefficient of the x term, by 2 to get -\frac{3}{16}. Then add the square of -\frac{3}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{8}x+\left(-\frac{3}{16}\right)^{2}=\left(-\frac{3}{16}\right)^{2}$$

Square -\frac{3}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{8}x+\frac{9}{256}=\frac{9}{256}$$

Factor x^{2}-\frac{3}{8}x+\frac{9}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{16}\right)^{2}=\frac{9}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{16}\right)^{2}}=\sqrt{\frac{9}{256}}$$

Simplify.

$$x-\frac{3}{16}=\frac{3}{16} x-\frac{3}{16}=-\frac{3}{16}$$

Add \frac{3}{16} to both sides of the equation.

$$x=\frac{3}{8} x=0$$