Solve left(1.25xright)^2-0.14x-22.92=0

$${ \left(1.25x \right) }^{ 2 } -0.14x-22.92 = 0$$

Solve for x

$$x = \frac{4 \sqrt{358174} + 28}{625} \approx 3.875048953$$

$$x=\frac{28-4\sqrt{358174}}{625}\approx -3.785448953$$

Steps Using the Quadratic Formula

Expand \left(1.25x\right)^{2}.

$$1.25^{2}x^{2}-0.14x-22.92=0$$

Calculate 1.25 to the power of 2 and get 1.5625.

$$1.5625x^{2}-0.14x-22.92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.5625 for a, -0.14 for b, and -22.92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.14\right)±\sqrt{\left(-0.14\right)^{2}-4\times 1.5625\left(-22.92\right)}}{2\times 1.5625}$$

Square -0.14 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.0196-4\times 1.5625\left(-22.92\right)}}{2\times 1.5625}$$

Multiply -4 times 1.5625.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.0196-6.25\left(-22.92\right)}}{2\times 1.5625}$$

Multiply -6.25 times -22.92 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.0196+143.25}}{2\times 1.5625}$$

Add 0.0196 to 143.25 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.14\right)±\sqrt{143.2696}}{2\times 1.5625}$$

Take the square root of 143.2696.

$$x=\frac{-\left(-0.14\right)±\frac{\sqrt{358174}}{50}}{2\times 1.5625}$$

The opposite of -0.14 is 0.14.

$$x=\frac{0.14±\frac{\sqrt{358174}}{50}}{2\times 1.5625}$$

Multiply 2 times 1.5625.

$$x=\frac{0.14±\frac{\sqrt{358174}}{50}}{3.125}$$

Now solve the equation x=\frac{0.14±\frac{\sqrt{358174}}{50}}{3.125} when ± is plus. Add 0.14 to \frac{\sqrt{358174}}{50}.

$$x=\frac{\sqrt{358174}+7}{3.125\times 50}$$

Divide \frac{7+\sqrt{358174}}{50} by 3.125 by multiplying \frac{7+\sqrt{358174}}{50} by the reciprocal of 3.125.

$$x=\frac{4\sqrt{358174}+28}{625}$$

Now solve the equation x=\frac{0.14±\frac{\sqrt{358174}}{50}}{3.125} when ± is minus. Subtract \frac{\sqrt{358174}}{50} from 0.14.

$$x=\frac{7-\sqrt{358174}}{3.125\times 50}$$

Divide \frac{7-\sqrt{358174}}{50} by 3.125 by multiplying \frac{7-\sqrt{358174}}{50} by the reciprocal of 3.125.

$$x=\frac{28-4\sqrt{358174}}{625}$$

The equation is now solved.

$$x=\frac{4\sqrt{358174}+28}{625} x=\frac{28-4\sqrt{358174}}{625}$$

Steps for Completing the Square

Expand \left(1.25x\right)^{2}.

$$1.25^{2}x^{2}-0.14x-22.92=0$$

Calculate 1.25 to the power of 2 and get 1.5625.

$$1.5625x^{2}-0.14x-22.92=0$$

Add 22.92 to both sides. Anything plus zero gives itself.

$$1.5625x^{2}-0.14x=22.92$$

Divide both sides of the equation by 1.5625, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.5625x^{2}-0.14x}{1.5625}=\frac{22.92}{1.5625}$$

Dividing by 1.5625 undoes the multiplication by 1.5625.

$$x^{2}+\left(-\frac{0.14}{1.5625}\right)x=\frac{22.92}{1.5625}$$

Divide -0.14 by 1.5625 by multiplying -0.14 by the reciprocal of 1.5625.

$$x^{2}-0.0896x=\frac{22.92}{1.5625}$$

Divide 22.92 by 1.5625 by multiplying 22.92 by the reciprocal of 1.5625.

$$x^{2}-0.0896x=14.6688$$

Divide -0.0896, the coefficient of the x term, by 2 to get -0.0448. Then add the square of -0.0448 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.0896x+\left(-0.0448\right)^{2}=14.6688+\left(-0.0448\right)^{2}$$

Square -0.0448 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.0896x+0.00200704=14.6688+0.00200704$$

Add 14.6688 to 0.00200704 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.0896x+0.00200704=14.67080704$$

Factor x^{2}-0.0896x+0.00200704. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.0448\right)^{2}=14.67080704$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.0448\right)^{2}}=\sqrt{14.67080704}$$

Simplify.

$$x-0.0448=\frac{4\sqrt{358174}}{625} x-0.0448=-\frac{4\sqrt{358174}}{625}$$

Add 0.0448 to both sides of the equation.

$$x=\frac{4\sqrt{358174}+28}{625} x=\frac{28-4\sqrt{358174}}{625}$$