Solve x^2+70x+1209=0

$${ x }^{ 2 } +70x+1209=0$$

Solve for x

$$x=-39$$

$$x=-31$$

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.

$$x^{2}+70x+1209=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 70 for b, and 1209 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\times 1209}}{2}$$

Square 70.

$$x=\frac{-70±\sqrt{4900-4\times 1209}}{2}$$

Multiply -4 times 1209.

$$x=\frac{-70±\sqrt{4900-4836}}{2}$$

Add 4900 to -4836.

$$x=\frac{-70±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-70±8}{2}$$

Now solve the equation x=\frac{-70±8}{2} when ± is plus. Add -70 to 8.

$$x=-\frac{62}{2}$$

Divide -62 by 2.

$$x=-31$$

Now solve the equation x=\frac{-70±8}{2} when ± is minus. Subtract 8 from -70.

$$x=-\frac{78}{2}$$

Divide -78 by 2.

$$x=-39$$

The equation is now solved.

$$x=-31 x=-39$$

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.

$$x^{2}+70x+1209=0$$

Subtract 1209 from both sides of the equation.

$$x^{2}+70x+1209-1209=-1209$$

Subtracting 1209 from itself leaves 0.

$$x^{2}+70x=-1209$$

Divide 70, the coefficient of the x term, by 2 to get 35. Then add the square of 35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+70x+35^{2}=-1209+35^{2}$$

Square 35.

$$x^{2}+70x+1225=-1209+1225$$

Add -1209 to 1225.

$$x^{2}+70x+1225=16$$

Factor x^{2}+70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+35\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+35\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+35=4 x+35=-4$$

Subtract 35 from both sides of the equation.

$$x=-31 x=-39$$