Solve 5=0.014x^2-1.18x+28.23

$$5 = 0.014 x ^ { 2 } - 1.18 x + 28.23$$

Solve for x

$$x = \frac{2 \sqrt{1430} + 295}{7} \approx 52.947240229$$

$$x = \frac{295 - 2 \sqrt{1430}}{7} \approx 31.338474056$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$0.014x^{2}-1.18x+28.23=5$$

Subtract 5 from both sides.

$$0.014x^{2}-1.18x+28.23-5=0$$

Subtract 5 from 28.23 to get 23.23.

$$0.014x^{2}-1.18x+23.23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.014 for a, -1.18 for b, and 23.23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.18\right)±\sqrt{\left(-1.18\right)^{2}-4\times 0.014\times 23.23}}{2\times 0.014}$$

Square -1.18 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.18\right)±\sqrt{1.3924-4\times 0.014\times 23.23}}{2\times 0.014}$$

Multiply -4 times 0.014.

$$x=\frac{-\left(-1.18\right)±\sqrt{1.3924-0.056\times 23.23}}{2\times 0.014}$$

Multiply -0.056 times 23.23 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.18\right)±\sqrt{1.3924-1.30088}}{2\times 0.014}$$

Add 1.3924 to -1.30088 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.18\right)±\sqrt{0.09152}}{2\times 0.014}$$

Take the square root of 0.09152.

$$x=\frac{-\left(-1.18\right)±\frac{\sqrt{1430}}{125}}{2\times 0.014}$$

The opposite of -1.18 is 1.18.

$$x=\frac{1.18±\frac{\sqrt{1430}}{125}}{2\times 0.014}$$

Multiply 2 times 0.014.

$$x=\frac{1.18±\frac{\sqrt{1430}}{125}}{0.028}$$

Now solve the equation x=\frac{1.18±\frac{\sqrt{1430}}{125}}{0.028} when ± is plus. Add 1.18 to \frac{\sqrt{1430}}{125}.

$$x=\frac{\frac{\sqrt{1430}}{125}+\frac{59}{50}}{0.028}$$

Divide \frac{59}{50}+\frac{\sqrt{1430}}{125} by 0.028 by multiplying \frac{59}{50}+\frac{\sqrt{1430}}{125} by the reciprocal of 0.028.

$$x=\frac{2\sqrt{1430}+295}{7}$$

Now solve the equation x=\frac{1.18±\frac{\sqrt{1430}}{125}}{0.028} when ± is minus. Subtract \frac{\sqrt{1430}}{125} from 1.18.

$$x=\frac{-\frac{\sqrt{1430}}{125}+\frac{59}{50}}{0.028}$$

Divide \frac{59}{50}-\frac{\sqrt{1430}}{125} by 0.028 by multiplying \frac{59}{50}-\frac{\sqrt{1430}}{125} by the reciprocal of 0.028.

$$x=\frac{295-2\sqrt{1430}}{7}$$

The equation is now solved.

$$x=\frac{2\sqrt{1430}+295}{7} x=\frac{295-2\sqrt{1430}}{7}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$0.014x^{2}-1.18x+28.23=5$$

Subtract 28.23 from both sides.

$$0.014x^{2}-1.18x=5-28.23$$

Subtract 28.23 from 5 to get -23.23.

$$0.014x^{2}-1.18x=-23.23$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$0.014x^{2}-1.18x=-\frac{2323}{100}$$

Divide both sides of the equation by 0.014, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.014x^{2}-1.18x}{0.014}=-\frac{\frac{2323}{100}}{0.014}$$

Dividing by 0.014 undoes the multiplication by 0.014.

$$x^{2}+\left(-\frac{1.18}{0.014}\right)x=-\frac{\frac{2323}{100}}{0.014}$$

Divide -1.18 by 0.014 by multiplying -1.18 by the reciprocal of 0.014.

$$x^{2}-\frac{590}{7}x=-\frac{\frac{2323}{100}}{0.014}$$

Divide -\frac{2323}{100} by 0.014 by multiplying -\frac{2323}{100} by the reciprocal of 0.014.

$$x^{2}-\frac{590}{7}x=-\frac{11615}{7}$$

Divide -\frac{590}{7}, the coefficient of the x term, by 2 to get -\frac{295}{7}. Then add the square of -\frac{295}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{590}{7}x+\left(-\frac{295}{7}\right)^{2}=-\frac{11615}{7}+\left(-\frac{295}{7}\right)^{2}$$

Square -\frac{295}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{590}{7}x+\frac{87025}{49}=-\frac{11615}{7}+\frac{87025}{49}$$

Add -\frac{11615}{7} to \frac{87025}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{590}{7}x+\frac{87025}{49}=\frac{5720}{49}$$

Factor x^{2}-\frac{590}{7}x+\frac{87025}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{295}{7}\right)^{2}=\frac{5720}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{295}{7}\right)^{2}}=\sqrt{\frac{5720}{49}}$$

Simplify.

$$x-\frac{295}{7}=\frac{2\sqrt{1430}}{7} x-\frac{295}{7}=-\frac{2\sqrt{1430}}{7}$$

Add \frac{295}{7} to both sides of the equation.

$$x=\frac{2\sqrt{1430}+295}{7} x=\frac{295-2\sqrt{1430}}{7}$$