Solve 4p^2+120p=-124

$$4 p ^ { 2 } + 120 p = - 124$$

Solve for p (complex solution)

$$p=\sqrt{194}-15\approx -1.071611723$$

$$p=-\left(\sqrt{194}+15\right)\approx -28.928388277$$

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.

$$4p^{2}+120p=-124$$

Add 124 to both sides of the equation.

$$4p^{2}+120p-\left(-124\right)=-124-\left(-124\right)$$

Subtracting -124 from itself leaves 0.

$$4p^{2}+120p-\left(-124\right)=0$$

Subtract -124 from 0.

$$4p^{2}+120p+124=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 120 for b, and 124 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-120±\sqrt{120^{2}-4\times 4\times 124}}{2\times 4}$$

Square 120.

$$p=\frac{-120±\sqrt{14400-4\times 4\times 124}}{2\times 4}$$

Multiply -4 times 4.

$$p=\frac{-120±\sqrt{14400-16\times 124}}{2\times 4}$$

Multiply -16 times 124.

$$p=\frac{-120±\sqrt{14400-1984}}{2\times 4}$$

Add 14400 to -1984.

$$p=\frac{-120±\sqrt{12416}}{2\times 4}$$

Take the square root of 12416.

$$p=\frac{-120±8\sqrt{194}}{2\times 4}$$

Multiply 2 times 4.

$$p=\frac{-120±8\sqrt{194}}{8}$$

Now solve the equation p=\frac{-120±8\sqrt{194}}{8} when ± is plus. Add -120 to 8\sqrt{194}.

$$p=\frac{8\sqrt{194}-120}{8}$$

Divide -120+8\sqrt{194} by 8.

$$p=\sqrt{194}-15$$

Now solve the equation p=\frac{-120±8\sqrt{194}}{8} when ± is minus. Subtract 8\sqrt{194} from -120.

$$p=\frac{-8\sqrt{194}-120}{8}$$

Divide -120-8\sqrt{194} by 8.

$$p=-\sqrt{194}-15$$

The equation is now solved.

$$p=\sqrt{194}-15 p=-\sqrt{194}-15$$

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.

$$4p^{2}+120p=-124$$

Divide both sides by 4.

$$\frac{4p^{2}+120p}{4}=-\frac{124}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$p^{2}+\frac{120}{4}p=-\frac{124}{4}$$

Divide 120 by 4.

$$p^{2}+30p=-\frac{124}{4}$$

Divide -124 by 4.

$$p^{2}+30p=-31$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+30p+15^{2}=-31+15^{2}$$

Square 15.

$$p^{2}+30p+225=-31+225$$

Add -31 to 225.

$$p^{2}+30p+225=194$$

Factor p^{2}+30p+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+15\right)^{2}=194$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+15\right)^{2}}=\sqrt{194}$$

Simplify.

$$p+15=\sqrt{194} p+15=-\sqrt{194}$$

Subtract 15 from both sides of the equation.

$$p=\sqrt{194}-15 p=-\sqrt{194}-15$$

Solve for p

$$p=\sqrt{194}-15\approx -1.071611723$$

$$p=-\sqrt{194}-15\approx -28.928388277$$

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.

$$4p^{2}+120p=-124$$

Add 124 to both sides of the equation.

$$4p^{2}+120p-\left(-124\right)=-124-\left(-124\right)$$

Subtracting -124 from itself leaves 0.

$$4p^{2}+120p-\left(-124\right)=0$$

Subtract -124 from 0.

$$4p^{2}+120p+124=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 120 for b, and 124 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-120±\sqrt{120^{2}-4\times 4\times 124}}{2\times 4}$$

Square 120.

$$p=\frac{-120±\sqrt{14400-4\times 4\times 124}}{2\times 4}$$

Multiply -4 times 4.

$$p=\frac{-120±\sqrt{14400-16\times 124}}{2\times 4}$$

Multiply -16 times 124.

$$p=\frac{-120±\sqrt{14400-1984}}{2\times 4}$$

Add 14400 to -1984.

$$p=\frac{-120±\sqrt{12416}}{2\times 4}$$

Take the square root of 12416.

$$p=\frac{-120±8\sqrt{194}}{2\times 4}$$

Multiply 2 times 4.

$$p=\frac{-120±8\sqrt{194}}{8}$$

Now solve the equation p=\frac{-120±8\sqrt{194}}{8} when ± is plus. Add -120 to 8\sqrt{194}.

$$p=\frac{8\sqrt{194}-120}{8}$$

Divide -120+8\sqrt{194} by 8.

$$p=\sqrt{194}-15$$

Now solve the equation p=\frac{-120±8\sqrt{194}}{8} when ± is minus. Subtract 8\sqrt{194} from -120.

$$p=\frac{-8\sqrt{194}-120}{8}$$

Divide -120-8\sqrt{194} by 8.

$$p=-\sqrt{194}-15$$

The equation is now solved.

$$p=\sqrt{194}-15 p=-\sqrt{194}-15$$

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.

$$4p^{2}+120p=-124$$

Divide both sides by 4.

$$\frac{4p^{2}+120p}{4}=-\frac{124}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$p^{2}+\frac{120}{4}p=-\frac{124}{4}$$

Divide 120 by 4.

$$p^{2}+30p=-\frac{124}{4}$$

Divide -124 by 4.

$$p^{2}+30p=-31$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+30p+15^{2}=-31+15^{2}$$

Square 15.

$$p^{2}+30p+225=-31+225$$

Add -31 to 225.

$$p^{2}+30p+225=194$$

Factor p^{2}+30p+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+15\right)^{2}=194$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+15\right)^{2}}=\sqrt{194}$$

Simplify.

$$p+15=\sqrt{194} p+15=-\sqrt{194}$$

Subtract 15 from both sides of the equation.

$$p=\sqrt{194}-15 p=-\sqrt{194}-15$$