Solve x^2+(x-1)^2=74

$$x ^ { 2 } + ( x - 1 ) ^ { 2 } = 74$$

Solve for x

$$x = \frac{7 \sqrt{3} + 1}{2} \approx 6.562177826$$

$$x=\frac{1-7\sqrt{3}}{2}\approx -5.562177826$$

Steps Using the Quadratic Formula

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

$$x^{2}+x^{2}-2x+1=74$$

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

$$2x^{2}-2x+1=74$$

Subtract 74 from both sides.

$$2x^{2}-2x+1-74=0$$

Subtract 74 from 1 to get -73.

$$2x^{2}-2x-73=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and -73 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-73\right)}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-73\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-73\right)}}{2\times 2}$$

Multiply -8 times -73.

$$x=\frac{-\left(-2\right)±\sqrt{4+584}}{2\times 2}$$

Add 4 to 584.

$$x=\frac{-\left(-2\right)±\sqrt{588}}{2\times 2}$$

Take the square root of 588.

$$x=\frac{-\left(-2\right)±14\sqrt{3}}{2\times 2}$$

The opposite of -2 is 2.

$$x=\frac{2±14\sqrt{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±14\sqrt{3}}{4}$$

Now solve the equation x=\frac{2±14\sqrt{3}}{4} when ± is plus. Add 2 to 14\sqrt{3}.

$$x=\frac{14\sqrt{3}+2}{4}$$

Divide 2+14\sqrt{3} by 4.

$$x=\frac{7\sqrt{3}+1}{2}$$

Now solve the equation x=\frac{2±14\sqrt{3}}{4} when ± is minus. Subtract 14\sqrt{3} from 2.

$$x=\frac{2-14\sqrt{3}}{4}$$

Divide 2-14\sqrt{3} by 4.

$$x=\frac{1-7\sqrt{3}}{2}$$

The equation is now solved.

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

Steps for Completing the Square

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

$$x^{2}+x^{2}-2x+1=74$$

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

$$2x^{2}-2x+1=74$$

Subtract 1 from both sides.

$$2x^{2}-2x=74-1$$

Subtract 1 from 74 to get 73.

$$2x^{2}-2x=73$$

Divide both sides by 2.

$$\frac{2x^{2}-2x}{2}=\frac{73}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{2}{2}\right)x=\frac{73}{2}$$

Divide -2 by 2.

$$x^{2}-x=\frac{73}{2}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{73}{2}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=\frac{73}{2}+\frac{1}{4}$$

Add \frac{73}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{147}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{147}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{147}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{7\sqrt{3}}{2} x-\frac{1}{2}=-\frac{7\sqrt{3}}{2}$$

Add \frac{1}{2} to both sides of the equation.

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