Solve 5.975x^2+4501.25x-706653.125=0

$$5.975 x ^ { 2 } + 4501.25 x - 706653.125 = 0$$

Solve for x

$$x = \frac{10 \sqrt{148601045} - 90025}{239} \approx 133.376687385$$

$$x=\frac{-10\sqrt{148601045}-90025}{239}\approx -886.72396772$$

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.

$$5.975x^{2}+4501.25x-706653.125=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5.975 for a, 4501.25 for b, and -706653.125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4501.25±\sqrt{4501.25^{2}-4\times 5.975\left(-706653.125\right)}}{2\times 5.975}$$

Square 4501.25 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-4501.25±\sqrt{20261251.5625-4\times 5.975\left(-706653.125\right)}}{2\times 5.975}$$

Multiply -4 times 5.975.

$$x=\frac{-4501.25±\sqrt{20261251.5625-23.9\left(-706653.125\right)}}{2\times 5.975}$$

Multiply -23.9 times -706653.125 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-4501.25±\sqrt{\frac{324180025+270224155}{16}}}{2\times 5.975}$$

Add 20261251.5625 to 16889009.6875 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-4501.25±\sqrt{37150261.25}}{2\times 5.975}$$

Take the square root of 37150261.25.

$$x=\frac{-4501.25±\frac{\sqrt{148601045}}{2}}{2\times 5.975}$$

Multiply 2 times 5.975.

$$x=\frac{-4501.25±\frac{\sqrt{148601045}}{2}}{11.95}$$

Now solve the equation x=\frac{-4501.25±\frac{\sqrt{148601045}}{2}}{11.95} when ± is plus. Add -4501.25 to \frac{\sqrt{148601045}}{2}.

$$x=\frac{\frac{\sqrt{148601045}}{2}-\frac{18005}{4}}{11.95}$$

Divide -\frac{18005}{4}+\frac{\sqrt{148601045}}{2} by 11.95 by multiplying -\frac{18005}{4}+\frac{\sqrt{148601045}}{2} by the reciprocal of 11.95.

$$x=\frac{10\sqrt{148601045}-90025}{239}$$

Now solve the equation x=\frac{-4501.25±\frac{\sqrt{148601045}}{2}}{11.95} when ± is minus. Subtract \frac{\sqrt{148601045}}{2} from -4501.25.

$$x=\frac{-\frac{\sqrt{148601045}}{2}-\frac{18005}{4}}{11.95}$$

Divide -\frac{18005}{4}-\frac{\sqrt{148601045}}{2} by 11.95 by multiplying -\frac{18005}{4}-\frac{\sqrt{148601045}}{2} by the reciprocal of 11.95.

$$x=\frac{-10\sqrt{148601045}-90025}{239}$$

The equation is now solved.

$$x=\frac{10\sqrt{148601045}-90025}{239} x=\frac{-10\sqrt{148601045}-90025}{239}$$

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.

$$5.975x^{2}+4501.25x-706653.125=0$$

Add 706653.125 to both sides of the equation.

$$5.975x^{2}+4501.25x-706653.125-\left(-706653.125\right)=-\left(-706653.125\right)$$

Subtracting -706653.125 from itself leaves 0.

$$5.975x^{2}+4501.25x=-\left(-706653.125\right)$$

Subtract -706653.125 from 0.

$$5.975x^{2}+4501.25x=706653.125$$

Divide both sides of the equation by 5.975, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{5.975x^{2}+4501.25x}{5.975}=\frac{706653.125}{5.975}$$

Dividing by 5.975 undoes the multiplication by 5.975.

$$x^{2}+\frac{4501.25}{5.975}x=\frac{706653.125}{5.975}$$

Divide 4501.25 by 5.975 by multiplying 4501.25 by the reciprocal of 5.975.

$$x^{2}+\frac{180050}{239}x=\frac{706653.125}{5.975}$$

Divide 706653.125 by 5.975 by multiplying 706653.125 by the reciprocal of 5.975.

$$x^{2}+\frac{180050}{239}x=\frac{28266125}{239}$$

Divide \frac{180050}{239}, the coefficient of the x term, by 2 to get \frac{90025}{239}. Then add the square of \frac{90025}{239} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{180050}{239}x+\frac{90025}{239}^{2}=\frac{28266125}{239}+\frac{90025}{239}^{2}$$

Square \frac{90025}{239} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{180050}{239}x+\frac{8104500625}{57121}=\frac{28266125}{239}+\frac{8104500625}{57121}$$

Add \frac{28266125}{239} to \frac{8104500625}{57121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{180050}{239}x+\frac{8104500625}{57121}=\frac{14860104500}{57121}$$

Factor x^{2}+\frac{180050}{239}x+\frac{8104500625}{57121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{90025}{239}\right)^{2}=\frac{14860104500}{57121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{90025}{239}\right)^{2}}=\sqrt{\frac{14860104500}{57121}}$$

Simplify.

$$x+\frac{90025}{239}=\frac{10\sqrt{148601045}}{239} x+\frac{90025}{239}=-\frac{10\sqrt{148601045}}{239}$$

Subtract \frac{90025}{239} from both sides of the equation.

$$x=\frac{10\sqrt{148601045}-90025}{239} x=\frac{-10\sqrt{148601045}-90025}{239}$$