Solve 15.30quadx^2-30x-470=0

$$15.30 \quad x ^ { 2 } - 30 x - 470 = 0$$

Solve for x

$$x = \frac{20 \sqrt{206} + 50}{51} \approx 6.608901998$$

$$x=\frac{50-20\sqrt{206}}{51}\approx -4.648117684$$

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.

$$15.3x^{2}-30x-470=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15.3 for a, -30 for b, and -470 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 15.3\left(-470\right)}}{2\times 15.3}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 15.3\left(-470\right)}}{2\times 15.3}$$

Multiply -4 times 15.3.

$$x=\frac{-\left(-30\right)±\sqrt{900-61.2\left(-470\right)}}{2\times 15.3}$$

Multiply -61.2 times -470.

$$x=\frac{-\left(-30\right)±\sqrt{900+28764}}{2\times 15.3}$$

Add 900 to 28764.

$$x=\frac{-\left(-30\right)±\sqrt{29664}}{2\times 15.3}$$

Take the square root of 29664.

$$x=\frac{-\left(-30\right)±12\sqrt{206}}{2\times 15.3}$$

The opposite of -30 is 30.

$$x=\frac{30±12\sqrt{206}}{2\times 15.3}$$

Multiply 2 times 15.3.

$$x=\frac{30±12\sqrt{206}}{30.6}$$

Now solve the equation x=\frac{30±12\sqrt{206}}{30.6} when ± is plus. Add 30 to 12\sqrt{206}.

$$x=\frac{12\sqrt{206}+30}{30.6}$$

Divide 30+12\sqrt{206} by 30.6 by multiplying 30+12\sqrt{206} by the reciprocal of 30.6.

$$x=\frac{20\sqrt{206}+50}{51}$$

Now solve the equation x=\frac{30±12\sqrt{206}}{30.6} when ± is minus. Subtract 12\sqrt{206} from 30.

$$x=\frac{30-12\sqrt{206}}{30.6}$$

Divide 30-12\sqrt{206} by 30.6 by multiplying 30-12\sqrt{206} by the reciprocal of 30.6.

$$x=\frac{50-20\sqrt{206}}{51}$$

The equation is now solved.

$$x=\frac{20\sqrt{206}+50}{51} x=\frac{50-20\sqrt{206}}{51}$$

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.

$$15.3x^{2}-30x-470=0$$

Add 470 to both sides of the equation.

$$15.3x^{2}-30x-470-\left(-470\right)=-\left(-470\right)$$

Subtracting -470 from itself leaves 0.

$$15.3x^{2}-30x=-\left(-470\right)$$

Subtract -470 from 0.

$$15.3x^{2}-30x=470$$

Divide both sides of the equation by 15.3, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{15.3x^{2}-30x}{15.3}=\frac{470}{15.3}$$

Dividing by 15.3 undoes the multiplication by 15.3.

$$x^{2}+\left(-\frac{30}{15.3}\right)x=\frac{470}{15.3}$$

Divide -30 by 15.3 by multiplying -30 by the reciprocal of 15.3.

$$x^{2}-\frac{100}{51}x=\frac{470}{15.3}$$

Divide 470 by 15.3 by multiplying 470 by the reciprocal of 15.3.

$$x^{2}-\frac{100}{51}x=\frac{4700}{153}$$

Divide -\frac{100}{51}, the coefficient of the x term, by 2 to get -\frac{50}{51}. Then add the square of -\frac{50}{51} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{100}{51}x+\left(-\frac{50}{51}\right)^{2}=\frac{4700}{153}+\left(-\frac{50}{51}\right)^{2}$$

Square -\frac{50}{51} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{100}{51}x+\frac{2500}{2601}=\frac{4700}{153}+\frac{2500}{2601}$$

Add \frac{4700}{153} to \frac{2500}{2601} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{100}{51}x+\frac{2500}{2601}=\frac{82400}{2601}$$

Factor x^{2}-\frac{100}{51}x+\frac{2500}{2601}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{50}{51}\right)^{2}=\frac{82400}{2601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{50}{51}\right)^{2}}=\sqrt{\frac{82400}{2601}}$$

Simplify.

$$x-\frac{50}{51}=\frac{20\sqrt{206}}{51} x-\frac{50}{51}=-\frac{20\sqrt{206}}{51}$$

Add \frac{50}{51} to both sides of the equation.

$$x=\frac{20\sqrt{206}+50}{51} x=\frac{50-20\sqrt{206}}{51}$$