Solve -x^2-10x=-56

$$- x ^ { 2 } - 10 x = - 56$$

Solve for x

$$x=-14$$

$$x=4$$

Steps Using Factoring By Grouping

Add 56 to both sides.

$$-x^{2}-10x+56=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+56. To find a and b, set up a system to be solved.

$$a+b=-10 ab=-56=-56$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -56.

$$1,-56 2,-28 4,-14 7,-8$$

Calculate the sum for each pair.

$$1-56=-55 2-28=-26 4-14=-10 7-8=-1$$

The solution is the pair that gives sum -10.

$$a=4 b=-14$$

Rewrite -x^{2}-10x+56 as \left(-x^{2}+4x\right)+\left(-14x+56\right).

$$\left(-x^{2}+4x\right)+\left(-14x+56\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(-x+4\right)+14\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(x+14\right)$$

To find equation solutions, solve -x+4=0 and x+14=0.

$$x=4 x=-14$$

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.

$$-x^{2}-10x=-56$$

Add 56 to both sides of the equation.

$$-x^{2}-10x-\left(-56\right)=-56-\left(-56\right)$$

Subtracting -56 from itself leaves 0.

$$-x^{2}-10x-\left(-56\right)=0$$

Subtract -56 from 0.

$$-x^{2}-10x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -10 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-1\right)\times 56}}{2\left(-1\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-1\right)\times 56}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-10\right)±\sqrt{100+4\times 56}}{2\left(-1\right)}$$

Multiply 4 times 56.

$$x=\frac{-\left(-10\right)±\sqrt{100+224}}{2\left(-1\right)}$$

Add 100 to 224.

$$x=\frac{-\left(-10\right)±\sqrt{324}}{2\left(-1\right)}$$

Take the square root of 324.

$$x=\frac{-\left(-10\right)±18}{2\left(-1\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±18}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{10±18}{-2}$$

Now solve the equation x=\frac{10±18}{-2} when ± is plus. Add 10 to 18.

$$x=\frac{28}{-2}$$

Divide 28 by -2.

$$x=-14$$

Now solve the equation x=\frac{10±18}{-2} when ± is minus. Subtract 18 from 10.

$$x=-\frac{8}{-2}$$

Divide -8 by -2.

$$x=4$$

The equation is now solved.

$$x=-14 x=4$$

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.

$$-x^{2}-10x=-56$$

Divide both sides by -1.

$$\frac{-x^{2}-10x}{-1}=-\frac{56}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{10}{-1}\right)x=-\frac{56}{-1}$$

Divide -10 by -1.

$$x^{2}+10x=-\frac{56}{-1}$$

Divide -56 by -1.

$$x^{2}+10x=56$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=56+5^{2}$$

Square 5.

$$x^{2}+10x+25=56+25$$

Add 56 to 25.

$$x^{2}+10x+25=81$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{81}$$

Simplify.

$$x+5=9 x+5=-9$$

Subtract 5 from both sides of the equation.

$$x=4 x=-14$$