Solve 99lambda^2+42lambda-7=0

$$99 \lambda ^ { 2 } + 42 \lambda - 7 = 0$$

Solve for λ

$$\lambda =\frac{\sqrt{14}}{11}-\frac{7}{33}\approx 0.128029459$$

$$\lambda =-\frac{\sqrt{14}}{11}-\frac{7}{33}\approx -0.552271884$$

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.

$$99\lambda ^{2}+42\lambda -7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 99 for a, 42 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-42±\sqrt{42^{2}-4\times 99\left(-7\right)}}{2\times 99}$$

Square 42.

$$\lambda =\frac{-42±\sqrt{1764-4\times 99\left(-7\right)}}{2\times 99}$$

Multiply -4 times 99.

$$\lambda =\frac{-42±\sqrt{1764-396\left(-7\right)}}{2\times 99}$$

Multiply -396 times -7.

$$\lambda =\frac{-42±\sqrt{1764+2772}}{2\times 99}$$

Add 1764 to 2772.

$$\lambda =\frac{-42±\sqrt{4536}}{2\times 99}$$

Take the square root of 4536.

$$\lambda =\frac{-42±18\sqrt{14}}{2\times 99}$$

Multiply 2 times 99.

$$\lambda =\frac{-42±18\sqrt{14}}{198}$$

Now solve the equation \lambda =\frac{-42±18\sqrt{14}}{198} when ± is plus. Add -42 to 18\sqrt{14}.

$$\lambda =\frac{18\sqrt{14}-42}{198}$$

Divide -42+18\sqrt{14} by 198.

$$\lambda =\frac{\sqrt{14}}{11}-\frac{7}{33}$$

Now solve the equation \lambda =\frac{-42±18\sqrt{14}}{198} when ± is minus. Subtract 18\sqrt{14} from -42.

$$\lambda =\frac{-18\sqrt{14}-42}{198}$$

Divide -42-18\sqrt{14} by 198.

$$\lambda =-\frac{\sqrt{14}}{11}-\frac{7}{33}$$

The equation is now solved.

$$\lambda =\frac{\sqrt{14}}{11}-\frac{7}{33} \lambda =-\frac{\sqrt{14}}{11}-\frac{7}{33}$$

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.

$$99\lambda ^{2}+42\lambda -7=0$$

Add 7 to both sides of the equation.

$$99\lambda ^{2}+42\lambda -7-\left(-7\right)=-\left(-7\right)$$

Subtracting -7 from itself leaves 0.

$$99\lambda ^{2}+42\lambda =-\left(-7\right)$$

Subtract -7 from 0.

$$99\lambda ^{2}+42\lambda =7$$

Divide both sides by 99.

$$\frac{99\lambda ^{2}+42\lambda }{99}=\frac{7}{99}$$

Dividing by 99 undoes the multiplication by 99.

$$\lambda ^{2}+\frac{42}{99}\lambda =\frac{7}{99}$$

Reduce the fraction \frac{42}{99} to lowest terms by extracting and canceling out 3.

$$\lambda ^{2}+\frac{14}{33}\lambda =\frac{7}{99}$$

Divide \frac{14}{33}, the coefficient of the x term, by 2 to get \frac{7}{33}. Then add the square of \frac{7}{33} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}+\frac{14}{33}\lambda +\left(\frac{7}{33}\right)^{2}=\frac{7}{99}+\left(\frac{7}{33}\right)^{2}$$

Square \frac{7}{33} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}+\frac{14}{33}\lambda +\frac{49}{1089}=\frac{7}{99}+\frac{49}{1089}$$

Add \frac{7}{99} to \frac{49}{1089} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\lambda ^{2}+\frac{14}{33}\lambda +\frac{49}{1089}=\frac{14}{121}$$

Factor \lambda ^{2}+\frac{14}{33}\lambda +\frac{49}{1089}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\lambda +\frac{7}{33}\right)^{2}=\frac{14}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda +\frac{7}{33}\right)^{2}}=\sqrt{\frac{14}{121}}$$

Simplify.

$$\lambda +\frac{7}{33}=\frac{\sqrt{14}}{11} \lambda +\frac{7}{33}=-\frac{\sqrt{14}}{11}$$

Subtract \frac{7}{33} from both sides of the equation.

$$\lambda =\frac{\sqrt{14}}{11}-\frac{7}{33} \lambda =-\frac{\sqrt{14}}{11}-\frac{7}{33}$$