Solve 0.0149x^2+8.134x=2624.67

$$0.0149 x ^ { 2 } + 8.134 x = 2624.67$$

Solve for x

$$x = \frac{20 \sqrt{13912018} - 40670}{149} \approx 227.70225068$$

$$x=\frac{-20\sqrt{13912018}-40670}{149}\approx -773.608290948$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$0.0149x^{2}+8.134x=2624.67$$

Subtract 2624.67 from both sides of the equation.

$$0.0149x^{2}+8.134x-2624.67=2624.67-2624.67$$

Subtracting 2624.67 from itself leaves 0.

$$0.0149x^{2}+8.134x-2624.67=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0149 for a, 8.134 for b, and -2624.67 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8.134±\sqrt{8.134^{2}-4\times 0.0149\left(-2624.67\right)}}{2\times 0.0149}$$

Square 8.134 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-8.134±\sqrt{66.161956-4\times 0.0149\left(-2624.67\right)}}{2\times 0.0149}$$

Multiply -4 times 0.0149.

$$x=\frac{-8.134±\sqrt{66.161956-0.0596\left(-2624.67\right)}}{2\times 0.0149}$$

Multiply -0.0596 times -2624.67 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-8.134±\sqrt{\frac{16540489+39107583}{250000}}}{2\times 0.0149}$$

Add 66.161956 to 156.430332 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-8.134±\sqrt{222.592288}}{2\times 0.0149}$$

Take the square root of 222.592288.

$$x=\frac{-8.134±\frac{\sqrt{13912018}}{250}}{2\times 0.0149}$$

Multiply 2 times 0.0149.

$$x=\frac{-8.134±\frac{\sqrt{13912018}}{250}}{0.0298}$$

Now solve the equation x=\frac{-8.134±\frac{\sqrt{13912018}}{250}}{0.0298} when ± is plus. Add -8.134 to \frac{\sqrt{13912018}}{250}.

$$x=\frac{\frac{\sqrt{13912018}}{250}-\frac{4067}{500}}{0.0298}$$

Divide -\frac{4067}{500}+\frac{\sqrt{13912018}}{250} by 0.0298 by multiplying -\frac{4067}{500}+\frac{\sqrt{13912018}}{250} by the reciprocal of 0.0298.

$$x=\frac{20\sqrt{13912018}-40670}{149}$$

Now solve the equation x=\frac{-8.134±\frac{\sqrt{13912018}}{250}}{0.0298} when ± is minus. Subtract \frac{\sqrt{13912018}}{250} from -8.134.

$$x=\frac{-\frac{\sqrt{13912018}}{250}-\frac{4067}{500}}{0.0298}$$

Divide -\frac{4067}{500}-\frac{\sqrt{13912018}}{250} by 0.0298 by multiplying -\frac{4067}{500}-\frac{\sqrt{13912018}}{250} by the reciprocal of 0.0298.

$$x=\frac{-20\sqrt{13912018}-40670}{149}$$

The equation is now solved.

$$x=\frac{20\sqrt{13912018}-40670}{149} x=\frac{-20\sqrt{13912018}-40670}{149}$$

Steps for Completing the Square

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.

$$0.0149x^{2}+8.134x=2624.67$$

Divide both sides of the equation by 0.0149, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0149x^{2}+8.134x}{0.0149}=\frac{2624.67}{0.0149}$$

Dividing by 0.0149 undoes the multiplication by 0.0149.

$$x^{2}+\frac{8.134}{0.0149}x=\frac{2624.67}{0.0149}$$

Divide 8.134 by 0.0149 by multiplying 8.134 by the reciprocal of 0.0149.

$$x^{2}+\frac{81340}{149}x=\frac{2624.67}{0.0149}$$

Divide 2624.67 by 0.0149 by multiplying 2624.67 by the reciprocal of 0.0149.

$$x^{2}+\frac{81340}{149}x=\frac{26246700}{149}$$

Divide \frac{81340}{149}, the coefficient of the x term, by 2 to get \frac{40670}{149}. Then add the square of \frac{40670}{149} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{81340}{149}x+\frac{40670}{149}^{2}=\frac{26246700}{149}+\frac{40670}{149}^{2}$$

Square \frac{40670}{149} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{81340}{149}x+\frac{1654048900}{22201}=\frac{26246700}{149}+\frac{1654048900}{22201}$$

Add \frac{26246700}{149} to \frac{1654048900}{22201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{81340}{149}x+\frac{1654048900}{22201}=\frac{5564807200}{22201}$$

Factor x^{2}+\frac{81340}{149}x+\frac{1654048900}{22201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{40670}{149}\right)^{2}=\frac{5564807200}{22201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{40670}{149}\right)^{2}}=\sqrt{\frac{5564807200}{22201}}$$

Simplify.

$$x+\frac{40670}{149}=\frac{20\sqrt{13912018}}{149} x+\frac{40670}{149}=-\frac{20\sqrt{13912018}}{149}$$

Subtract \frac{40670}{149} from both sides of the equation.

$$x=\frac{20\sqrt{13912018}-40670}{149} x=\frac{-20\sqrt{13912018}-40670}{149}$$