Solve 7a^2+336=-98a

$$7 a ^ { 2 } + 336 = - 98 a$$

Solve for a

$$a=-8$$

$$a=-6$$

Steps Using the Quadratic Formula

Add 98a to both sides.

$$7a^{2}+336+98a=0$$

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.

$$7a^{2}+98a+336=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 98 for b, and 336 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-98±\sqrt{98^{2}-4\times 7\times 336}}{2\times 7}$$

Square 98.

$$a=\frac{-98±\sqrt{9604-4\times 7\times 336}}{2\times 7}$$

Multiply -4 times 7.

$$a=\frac{-98±\sqrt{9604-28\times 336}}{2\times 7}$$

Multiply -28 times 336.

$$a=\frac{-98±\sqrt{9604-9408}}{2\times 7}$$

Add 9604 to -9408.

$$a=\frac{-98±\sqrt{196}}{2\times 7}$$

Take the square root of 196.

$$a=\frac{-98±14}{2\times 7}$$

Multiply 2 times 7.

$$a=\frac{-98±14}{14}$$

Now solve the equation a=\frac{-98±14}{14} when ± is plus. Add -98 to 14.

$$a=-\frac{84}{14}$$

Divide -84 by 14.

$$a=-6$$

Now solve the equation a=\frac{-98±14}{14} when ± is minus. Subtract 14 from -98.

$$a=-\frac{112}{14}$$

Divide -112 by 14.

$$a=-8$$

The equation is now solved.

$$a=-6 a=-8$$

Steps for Completing the Square

Add 98a to both sides.

$$7a^{2}+336+98a=0$$

Subtract 336 from both sides. Anything subtracted from zero gives its negation.

$$7a^{2}+98a=-336$$

Divide both sides by 7.

$$\frac{7a^{2}+98a}{7}=-\frac{336}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$a^{2}+\frac{98}{7}a=-\frac{336}{7}$$

Divide 98 by 7.

$$a^{2}+14a=-\frac{336}{7}$$

Divide -336 by 7.

$$a^{2}+14a=-48$$

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

$$a^{2}+14a+7^{2}=-48+7^{2}$$

Square 7.

$$a^{2}+14a+49=-48+49$$

Add -48 to 49.

$$a^{2}+14a+49=1$$

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

$$\left(a+7\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+7\right)^{2}}=\sqrt{1}$$

Simplify.

$$a+7=1 a+7=-1$$

Subtract 7 from both sides of the equation.

$$a=-6 a=-8$$