Solve (x-0.14)^2=1+0.14^2

$$( x - 0.14 ) ^ { 2 } = 1 + 0.14 ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{2549} + 7}{50} \approx 1.149752445$$

$$x=\frac{7-\sqrt{2549}}{50}\approx -0.869752445$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-0.14\right)^{2}.

$$x^{2}-0.28x+0.0196=1+0.14^{2}$$

Calculate 0.14 to the power of 2 and get 0.0196.

$$x^{2}-0.28x+0.0196=1+0.0196$$

Add 1 and 0.0196 to get 1.0196.

$$x^{2}-0.28x+0.0196=1.0196$$

Subtract 1.0196 from both sides.

$$x^{2}-0.28x+0.0196-1.0196=0$$

Subtract 1.0196 from 0.0196 to get -1.

$$x^{2}-0.28x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -0.28 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.28\right)±\sqrt{\left(-0.28\right)^{2}-4\left(-1\right)}}{2}$$

Square -0.28 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.28\right)±\sqrt{0.0784-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-\left(-0.28\right)±\sqrt{0.0784+4}}{2}$$

Add 0.0784 to 4.

$$x=\frac{-\left(-0.28\right)±\sqrt{4.0784}}{2}$$

Take the square root of 4.0784.

$$x=\frac{-\left(-0.28\right)±\frac{\sqrt{2549}}{25}}{2}$$

The opposite of -0.28 is 0.28.

$$x=\frac{0.28±\frac{\sqrt{2549}}{25}}{2}$$

Now solve the equation x=\frac{0.28±\frac{\sqrt{2549}}{25}}{2} when ± is plus. Add 0.28 to \frac{\sqrt{2549}}{25}.

$$x=\frac{\sqrt{2549}+7}{2\times 25}$$

Divide \frac{7+\sqrt{2549}}{25} by 2.

$$x=\frac{\sqrt{2549}+7}{50}$$

Now solve the equation x=\frac{0.28±\frac{\sqrt{2549}}{25}}{2} when ± is minus. Subtract \frac{\sqrt{2549}}{25} from 0.28.

$$x=\frac{7-\sqrt{2549}}{2\times 25}$$

Divide \frac{7-\sqrt{2549}}{25} by 2.

$$x=\frac{7-\sqrt{2549}}{50}$$

The equation is now solved.

$$x=\frac{\sqrt{2549}+7}{50} x=\frac{7-\sqrt{2549}}{50}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-0.14\right)^{2}.

$$x^{2}-0.28x+0.0196=1+0.14^{2}$$

Calculate 0.14 to the power of 2 and get 0.0196.

$$x^{2}-0.28x+0.0196=1+0.0196$$

Add 1 to 0.0196.

$$x^{2}-0.28x+0.0196=1.0196$$

Factor x^{2}-0.28x+0.0196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.14\right)^{2}=1.0196$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.14\right)^{2}}=\sqrt{1.0196}$$

Simplify.

$$x-0.14=\frac{\sqrt{2549}}{50} x-0.14=-\frac{\sqrt{2549}}{50}$$

Add 0.14 to both sides of the equation.

$$x=\frac{\sqrt{2549}+7}{50} x=\frac{7-\sqrt{2549}}{50}$$