Solve 84j=-7j^2

$$84 j = - 7 j ^ { 2 }$$

Solve for j

$$j=-12$$

$$j=0$$

Steps Using Factoring

Add 7j^{2} to both sides.

$$84j+7j^{2}=0$$

Factor out j.

$$j\left(84+7j\right)=0$$

To find equation solutions, solve j=0 and 84+7j=0.

$$j=0 j=-12$$

Steps Using the Quadratic Formula

Add 7j^{2} to both sides.

$$84j+7j^{2}=0$$

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.

$$7j^{2}+84j=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 84 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$j=\frac{-84±\sqrt{84^{2}}}{2\times 7}$$

Take the square root of 84^{2}.

$$j=\frac{-84±84}{2\times 7}$$

Multiply 2 times 7.

$$j=\frac{-84±84}{14}$$

Now solve the equation j=\frac{-84±84}{14} when ± is plus. Add -84 to 84.

$$j=\frac{0}{14}$$

Divide 0 by 14.

$$j=0$$

Now solve the equation j=\frac{-84±84}{14} when ± is minus. Subtract 84 from -84.

$$j=-\frac{168}{14}$$

Divide -168 by 14.

$$j=-12$$

The equation is now solved.

$$j=0 j=-12$$

Steps for Completing the Square

Add 7j^{2} to both sides.

$$84j+7j^{2}=0$$

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.

$$7j^{2}+84j=0$$

Divide both sides by 7.

$$\frac{7j^{2}+84j}{7}=\frac{0}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$j^{2}+\frac{84}{7}j=\frac{0}{7}$$

Divide 84 by 7.

$$j^{2}+12j=\frac{0}{7}$$

Divide 0 by 7.

$$j^{2}+12j=0$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$j^{2}+12j+6^{2}=6^{2}$$

Square 6.

$$j^{2}+12j+36=36$$

Factor j^{2}+12j+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(j+6\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(j+6\right)^{2}}=\sqrt{36}$$

Simplify.

$$j+6=6 j+6=-6$$

Subtract 6 from both sides of the equation.

$$j=0 j=-12$$