Solve x^2+77x=-76

$$x ^ { 2 } + 77 x = - 76$$

Solve for x

$$x=-76$$

$$x=-1$$

Steps Using Factoring

Add 76 to both sides.

$$x^{2}+77x+76=0$$

To solve the equation, factor x^{2}+77x+76 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.

$$a+b=77 ab=76$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 76.

$$1,76 2,38 4,19$$

Calculate the sum for each pair.

$$1+76=77 2+38=40 4+19=23$$

The solution is the pair that gives sum 77.

$$a=1 b=76$$

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

$$\left(x+1\right)\left(x+76\right)$$

To find equation solutions, solve x+1=0 and x+76=0.

$$x=-1 x=-76$$

Steps Using Factoring By Grouping

Add 76 to both sides.

$$x^{2}+77x+76=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+76. To find a and b, set up a system to be solved.

$$a+b=77 ab=1\times 76=76$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 76.

$$1,76 2,38 4,19$$

Calculate the sum for each pair.

$$1+76=77 2+38=40 4+19=23$$

The solution is the pair that gives sum 77.

$$a=1 b=76$$

Rewrite x^{2}+77x+76 as \left(x^{2}+x\right)+\left(76x+76\right).

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

Factor out x in the first and 76 in the second group.

$$x\left(x+1\right)+76\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+76\right)$$

To find equation solutions, solve x+1=0 and x+76=0.

$$x=-1 x=-76$$

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.

$$x^{2}+77x=-76$$

Add 76 to both sides of the equation.

$$x^{2}+77x-\left(-76\right)=-76-\left(-76\right)$$

Subtracting -76 from itself leaves 0.

$$x^{2}+77x-\left(-76\right)=0$$

Subtract -76 from 0.

$$x^{2}+77x+76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 77 for b, and 76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-77±\sqrt{77^{2}-4\times 76}}{2}$$

Square 77.

$$x=\frac{-77±\sqrt{5929-4\times 76}}{2}$$

Multiply -4 times 76.

$$x=\frac{-77±\sqrt{5929-304}}{2}$$

Add 5929 to -304.

$$x=\frac{-77±\sqrt{5625}}{2}$$

Take the square root of 5625.

$$x=\frac{-77±75}{2}$$

Now solve the equation x=\frac{-77±75}{2} when ± is plus. Add -77 to 75.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-77±75}{2} when ± is minus. Subtract 75 from -77.

$$x=-\frac{152}{2}$$

Divide -152 by 2.

$$x=-76$$

The equation is now solved.

$$x=-1 x=-76$$

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.

$$x^{2}+77x=-76$$

Divide 77, the coefficient of the x term, by 2 to get \frac{77}{2}. Then add the square of \frac{77}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+77x+\left(\frac{77}{2}\right)^{2}=-76+\left(\frac{77}{2}\right)^{2}$$

Square \frac{77}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+77x+\frac{5929}{4}=-76+\frac{5929}{4}$$

Add -76 to \frac{5929}{4}.

$$x^{2}+77x+\frac{5929}{4}=\frac{5625}{4}$$

Factor x^{2}+77x+\frac{5929}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{77}{2}\right)^{2}=\frac{5625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{77}{2}\right)^{2}}=\sqrt{\frac{5625}{4}}$$

Simplify.

$$x+\frac{77}{2}=\frac{75}{2} x+\frac{77}{2}=-\frac{75}{2}$$

Subtract \frac{77}{2} from both sides of the equation.

$$x=-1 x=-76$$