Solve x^2+146x+4704=0

$${ x }^{ 2 } +146x+4704=0$$

Solve for x

$$x=-98$$

$$x=-48$$

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.

$$x^{2}+146x+4704=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 146 for b, and 4704 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-146±\sqrt{146^{2}-4\times 4704}}{2}$$

Square 146.

$$x=\frac{-146±\sqrt{21316-4\times 4704}}{2}$$

Multiply -4 times 4704.

$$x=\frac{-146±\sqrt{21316-18816}}{2}$$

Add 21316 to -18816.

$$x=\frac{-146±\sqrt{2500}}{2}$$

Take the square root of 2500.

$$x=\frac{-146±50}{2}$$

Now solve the equation x=\frac{-146±50}{2} when ± is plus. Add -146 to 50.

$$x=-\frac{96}{2}$$

Divide -96 by 2.

$$x=-48$$

Now solve the equation x=\frac{-146±50}{2} when ± is minus. Subtract 50 from -146.

$$x=-\frac{196}{2}$$

Divide -196 by 2.

$$x=-98$$

The equation is now solved.

$$x=-48 x=-98$$

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.

$$x^{2}+146x+4704=0$$

Subtract 4704 from both sides of the equation.

$$x^{2}+146x+4704-4704=-4704$$

Subtracting 4704 from itself leaves 0.

$$x^{2}+146x=-4704$$

Divide 146, the coefficient of the x term, by 2 to get 73. Then add the square of 73 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+146x+73^{2}=-4704+73^{2}$$

Square 73.

$$x^{2}+146x+5329=-4704+5329$$

Add -4704 to 5329.

$$x^{2}+146x+5329=625$$

Factor x^{2}+146x+5329. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+73\right)^{2}=625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+73\right)^{2}}=\sqrt{625}$$

Simplify.

$$x+73=25 x+73=-25$$

Subtract 73 from both sides of the equation.

$$x=-48 x=-98$$