Solve x^2-5x+3=3x-7

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

Solve for x

$$x=\sqrt{6}+4\approx 6.449489743$$

$$x=4-\sqrt{6}\approx 1.550510257$$

Steps Using the Quadratic Formula

Subtract 3x from both sides.

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

Combine -5x and -3x to get -8x.

$$x^{2}-8x+3=-7$$

Add 7 to both sides.

$$x^{2}-8x+3+7=0$$

Add 3 and 7 to get 10.

$$x^{2}-8x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 10}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-\left(-8\right)±\sqrt{64-40}}{2}$$

Add 64 to -40.

$$x=\frac{-\left(-8\right)±\sqrt{24}}{2}$$

Take the square root of 24.

$$x=\frac{-\left(-8\right)±2\sqrt{6}}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{6}}{2}$$

Now solve the equation x=\frac{8±2\sqrt{6}}{2} when ± is plus. Add 8 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}+8}{2}$$

Divide 8+2\sqrt{6} by 2.

$$x=\sqrt{6}+4$$

Now solve the equation x=\frac{8±2\sqrt{6}}{2} when ± is minus. Subtract 2\sqrt{6} from 8.

$$x=\frac{8-2\sqrt{6}}{2}$$

Divide 8-2\sqrt{6} by 2.

$$x=4-\sqrt{6}$$

The equation is now solved.

$$x=\sqrt{6}+4 x=4-\sqrt{6}$$

Steps for Completing the Square

Subtract 3x from both sides.

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

Combine -5x and -3x to get -8x.

$$x^{2}-8x+3=-7$$

Subtract 3 from both sides.

$$x^{2}-8x=-7-3$$

Subtract 3 from -7 to get -10.

$$x^{2}-8x=-10$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -4.

$$x^{2}-8x+16=-10+16$$

Add -10 to 16.

$$x^{2}-8x+16=6$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-4=\sqrt{6} x-4=-\sqrt{6}$$

Add 4 to both sides of the equation.

$$x=\sqrt{6}+4 x=4-\sqrt{6}$$