Solve x^2+76x-4332=0

$${ x }^{ 2 } +76x-4332=0$$

Solve for x

$$x=-114$$

$$x=38$$

Steps Using Factoring

To solve the equation, factor x^{2}+76x-4332 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=76 ab=-4332$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4332.

$$-1,4332 -2,2166 -3,1444 -4,1083 -6,722 -12,361 -19,228 -38,114 -57,76$$

Calculate the sum for each pair.

$$-1+4332=4331 -2+2166=2164 -3+1444=1441 -4+1083=1079 -6+722=716 -12+361=349 -19+228=209 -38+114=76 -57+76=19$$

The solution is the pair that gives sum 76.

$$a=-38 b=114$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-38\right)\left(x+114\right)$$

To find equation solutions, solve x-38=0 and x+114=0.

$$x=38 x=-114$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-4332. To find a and b, set up a system to be solved.

$$a+b=76 ab=1\left(-4332\right)=-4332$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4332.

$$-1,4332 -2,2166 -3,1444 -4,1083 -6,722 -12,361 -19,228 -38,114 -57,76$$

Calculate the sum for each pair.

$$-1+4332=4331 -2+2166=2164 -3+1444=1441 -4+1083=1079 -6+722=716 -12+361=349 -19+228=209 -38+114=76 -57+76=19$$

The solution is the pair that gives sum 76.

$$a=-38 b=114$$

Rewrite x^{2}+76x-4332 as \left(x^{2}-38x\right)+\left(114x-4332\right).

$$\left(x^{2}-38x\right)+\left(114x-4332\right)$$

Factor out x in the first and 114 in the second group.

$$x\left(x-38\right)+114\left(x-38\right)$$

Factor out common term x-38 by using distributive property.

$$\left(x-38\right)\left(x+114\right)$$

To find equation solutions, solve x-38=0 and x+114=0.

$$x=38 x=-114$$

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.

$$x^{2}+76x-4332=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 76 for b, and -4332 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-4332\right)}}{2}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-4332\right)}}{2}$$

Multiply -4 times -4332.

$$x=\frac{-76±\sqrt{5776+17328}}{2}$$

Add 5776 to 17328.

$$x=\frac{-76±\sqrt{23104}}{2}$$

Take the square root of 23104.

$$x=\frac{-76±152}{2}$$

Now solve the equation x=\frac{-76±152}{2} when ± is plus. Add -76 to 152.

$$x=\frac{76}{2}$$

Divide 76 by 2.

$$x=38$$

Now solve the equation x=\frac{-76±152}{2} when ± is minus. Subtract 152 from -76.

$$x=-\frac{228}{2}$$

Divide -228 by 2.

$$x=-114$$

The equation is now solved.

$$x=38 x=-114$$

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.

$$x^{2}+76x-4332=0$$

Add 4332 to both sides of the equation.

$$x^{2}+76x-4332-\left(-4332\right)=-\left(-4332\right)$$

Subtracting -4332 from itself leaves 0.

$$x^{2}+76x=-\left(-4332\right)$$

Subtract -4332 from 0.

$$x^{2}+76x=4332$$

Divide 76, the coefficient of the x term, by 2 to get 38. Then add the square of 38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+76x+38^{2}=4332+38^{2}$$

Square 38.

$$x^{2}+76x+1444=4332+1444$$

Add 4332 to 1444.

$$x^{2}+76x+1444=5776$$

Factor x^{2}+76x+1444. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+38\right)^{2}=5776$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+38\right)^{2}}=\sqrt{5776}$$

Simplify.

$$x+38=76 x+38=-76$$

Subtract 38 from both sides of the equation.

$$x=38 x=-114$$