Solve 1^3-29lambda^2+47lambda-60=0

$$1 ^ { 3 } - 29 \lambda ^ { 2 } + 47 \lambda - 60 = 0$$

Solve for λ

$$\lambda =\frac{-3\sqrt{515}i+47}{58}\approx 0.810344828-1.173807488i$$

$$\lambda =\frac{47+3\sqrt{515}i}{58}\approx 0.810344828+1.173807488i$$

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.

$$-29\lambda ^{2}+47\lambda -59=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -29 for a, 47 for b, and -59 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-47±\sqrt{47^{2}-4\left(-29\right)\left(-59\right)}}{2\left(-29\right)}$$

Square 47.

$$\lambda =\frac{-47±\sqrt{2209-4\left(-29\right)\left(-59\right)}}{2\left(-29\right)}$$

Multiply -4 times -29.

$$\lambda =\frac{-47±\sqrt{2209+116\left(-59\right)}}{2\left(-29\right)}$$

Multiply 116 times -59.

$$\lambda =\frac{-47±\sqrt{2209-6844}}{2\left(-29\right)}$$

Add 2209 to -6844.

$$\lambda =\frac{-47±\sqrt{-4635}}{2\left(-29\right)}$$

Take the square root of -4635.

$$\lambda =\frac{-47±3\sqrt{515}i}{2\left(-29\right)}$$

Multiply 2 times -29.

$$\lambda =\frac{-47±3\sqrt{515}i}{-58}$$

Now solve the equation \lambda =\frac{-47±3\sqrt{515}i}{-58} when ± is plus. Add -47 to 3i\sqrt{515}.

$$\lambda =\frac{-47+3\sqrt{515}i}{-58}$$

Divide -47+3i\sqrt{515} by -58.

$$\lambda =\frac{-3\sqrt{515}i+47}{58}$$

Now solve the equation \lambda =\frac{-47±3\sqrt{515}i}{-58} when ± is minus. Subtract 3i\sqrt{515} from -47.

$$\lambda =\frac{-3\sqrt{515}i-47}{-58}$$

Divide -47-3i\sqrt{515} by -58.

$$\lambda =\frac{47+3\sqrt{515}i}{58}$$

The equation is now solved.

$$\lambda =\frac{-3\sqrt{515}i+47}{58} \lambda =\frac{47+3\sqrt{515}i}{58}$$

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.

$$-29\lambda ^{2}+47\lambda -59=0$$

Add 59 to both sides of the equation.

$$-29\lambda ^{2}+47\lambda -59-\left(-59\right)=-\left(-59\right)$$

Subtracting -59 from itself leaves 0.

$$-29\lambda ^{2}+47\lambda =-\left(-59\right)$$

Subtract -59 from 0.

$$-29\lambda ^{2}+47\lambda =59$$

Divide both sides by -29.

$$\frac{-29\lambda ^{2}+47\lambda }{-29}=\frac{59}{-29}$$

Dividing by -29 undoes the multiplication by -29.

$$\lambda ^{2}+\frac{47}{-29}\lambda =\frac{59}{-29}$$

Divide 47 by -29.

$$\lambda ^{2}-\frac{47}{29}\lambda =\frac{59}{-29}$$

Divide 59 by -29.

$$\lambda ^{2}-\frac{47}{29}\lambda =-\frac{59}{29}$$

Divide -\frac{47}{29}, the coefficient of the x term, by 2 to get -\frac{47}{58}. Then add the square of -\frac{47}{58} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}-\frac{47}{29}\lambda +\left(-\frac{47}{58}\right)^{2}=-\frac{59}{29}+\left(-\frac{47}{58}\right)^{2}$$

Square -\frac{47}{58} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}-\frac{47}{29}\lambda +\frac{2209}{3364}=-\frac{59}{29}+\frac{2209}{3364}$$

Add -\frac{59}{29} to \frac{2209}{3364} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\lambda ^{2}-\frac{47}{29}\lambda +\frac{2209}{3364}=-\frac{4635}{3364}$$

Factor \lambda ^{2}-\frac{47}{29}\lambda +\frac{2209}{3364}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\lambda -\frac{47}{58}\right)^{2}=-\frac{4635}{3364}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{47}{58}\right)^{2}}=\sqrt{-\frac{4635}{3364}}$$

Simplify.

$$\lambda -\frac{47}{58}=\frac{3\sqrt{515}i}{58} \lambda -\frac{47}{58}=-\frac{3\sqrt{515}i}{58}$$

Add \frac{47}{58} to both sides of the equation.

$$\lambda =\frac{47+3\sqrt{515}i}{58} \lambda =\frac{-3\sqrt{515}i+47}{58}$$