Solve 1456.875+735x=29000x^2

$$1456.875 + 735 x = 29000 x ^ { 2 }$$

Solve for x

$$x=\frac{3\sqrt{753501}+147}{11600}\approx 0.237166643$$

$$x=\frac{147-3\sqrt{753501}}{11600}\approx -0.211821816$$

Steps Using the Quadratic Formula

Subtract 29000x^{2} from both sides.

$$1456.875+735x-29000x^{2}=0$$

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.

$$-29000x^{2}+735x+1456.875=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -29000 for a, 735 for b, and 1456.875 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-735±\sqrt{735^{2}-4\left(-29000\right)\times 1456.875}}{2\left(-29000\right)}$$

Square 735.

$$x=\frac{-735±\sqrt{540225-4\left(-29000\right)\times 1456.875}}{2\left(-29000\right)}$$

Multiply -4 times -29000.

$$x=\frac{-735±\sqrt{540225+116000\times 1456.875}}{2\left(-29000\right)}$$

Multiply 116000 times 1456.875.

$$x=\frac{-735±\sqrt{540225+168997500}}{2\left(-29000\right)}$$

Add 540225 to 168997500.

$$x=\frac{-735±\sqrt{169537725}}{2\left(-29000\right)}$$

Take the square root of 169537725.

$$x=\frac{-735±15\sqrt{753501}}{2\left(-29000\right)}$$

Multiply 2 times -29000.

$$x=\frac{-735±15\sqrt{753501}}{-58000}$$

Now solve the equation x=\frac{-735±15\sqrt{753501}}{-58000} when ± is plus. Add -735 to 15\sqrt{753501}.

$$x=\frac{15\sqrt{753501}-735}{-58000}$$

Divide -735+15\sqrt{753501} by -58000.

$$x=\frac{147-3\sqrt{753501}}{11600}$$

Now solve the equation x=\frac{-735±15\sqrt{753501}}{-58000} when ± is minus. Subtract 15\sqrt{753501} from -735.

$$x=\frac{-15\sqrt{753501}-735}{-58000}$$

Divide -735-15\sqrt{753501} by -58000.

$$x=\frac{3\sqrt{753501}+147}{11600}$$

The equation is now solved.

$$x=\frac{147-3\sqrt{753501}}{11600} x=\frac{3\sqrt{753501}+147}{11600}$$

Steps for Completing the Square

Subtract 29000x^{2} from both sides.

$$1456.875+735x-29000x^{2}=0$$

Subtract 1456.875 from both sides. Anything subtracted from zero gives its negation.

$$735x-29000x^{2}=-1456.875$$

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.

$$-29000x^{2}+735x=-1456.875$$

Divide both sides by -29000.

$$\frac{-29000x^{2}+735x}{-29000}=-\frac{1456.875}{-29000}$$

Dividing by -29000 undoes the multiplication by -29000.

$$x^{2}+\frac{735}{-29000}x=-\frac{1456.875}{-29000}$$

Reduce the fraction \frac{735}{-29000} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{147}{5800}x=-\frac{1456.875}{-29000}$$

Divide -1456.875 by -29000.

$$x^{2}-\frac{147}{5800}x=\frac{2331}{46400}$$

Divide -\frac{147}{5800}, the coefficient of the x term, by 2 to get -\frac{147}{11600}. Then add the square of -\frac{147}{11600} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{147}{5800}x+\left(-\frac{147}{11600}\right)^{2}=\frac{2331}{46400}+\left(-\frac{147}{11600}\right)^{2}$$

Square -\frac{147}{11600} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{147}{5800}x+\frac{21609}{134560000}=\frac{2331}{46400}+\frac{21609}{134560000}$$

Add \frac{2331}{46400} to \frac{21609}{134560000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{147}{5800}x+\frac{21609}{134560000}=\frac{6781509}{134560000}$$

Factor x^{2}-\frac{147}{5800}x+\frac{21609}{134560000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{147}{11600}\right)^{2}=\frac{6781509}{134560000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{147}{11600}\right)^{2}}=\sqrt{\frac{6781509}{134560000}}$$

Simplify.

$$x-\frac{147}{11600}=\frac{3\sqrt{753501}}{11600} x-\frac{147}{11600}=-\frac{3\sqrt{753501}}{11600}$$

Add \frac{147}{11600} to both sides of the equation.

$$x=\frac{3\sqrt{753501}+147}{11600} x=\frac{147-3\sqrt{753501}}{11600}$$