Solve 1034=x^2+30x-110

$$1034 = x ^ { 2 } + 30 x - 110$$

Solve for x

$$x=-52$$

$$x=22$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+30x-110=1034$$

Subtract 1034 from both sides.

$$x^{2}+30x-110-1034=0$$

Subtract 1034 from -110 to get -1144.

$$x^{2}+30x-1144=0$$

To solve the equation, factor x^{2}+30x-1144 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.

$$a+b=30 ab=-1144$$

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 -1144.

$$-1,1144 -2,572 -4,286 -8,143 -11,104 -13,88 -22,52 -26,44$$

Calculate the sum for each pair.

$$-1+1144=1143 -2+572=570 -4+286=282 -8+143=135 -11+104=93 -13+88=75 -22+52=30 -26+44=18$$

The solution is the pair that gives sum 30.

$$a=-22 b=52$$

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

$$\left(x-22\right)\left(x+52\right)$$

To find equation solutions, solve x-22=0 and x+52=0.

$$x=22 x=-52$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+30x-110=1034$$

Subtract 1034 from both sides.

$$x^{2}+30x-110-1034=0$$

Subtract 1034 from -110 to get -1144.

$$x^{2}+30x-1144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-1144. To find a and b, set up a system to be solved.

$$a+b=30 ab=1\left(-1144\right)=-1144$$

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 -1144.

$$-1,1144 -2,572 -4,286 -8,143 -11,104 -13,88 -22,52 -26,44$$

Calculate the sum for each pair.

$$-1+1144=1143 -2+572=570 -4+286=282 -8+143=135 -11+104=93 -13+88=75 -22+52=30 -26+44=18$$

The solution is the pair that gives sum 30.

$$a=-22 b=52$$

Rewrite x^{2}+30x-1144 as \left(x^{2}-22x\right)+\left(52x-1144\right).

$$\left(x^{2}-22x\right)+\left(52x-1144\right)$$

Factor out x in the first and 52 in the second group.

$$x\left(x-22\right)+52\left(x-22\right)$$

Factor out common term x-22 by using distributive property.

$$\left(x-22\right)\left(x+52\right)$$

To find equation solutions, solve x-22=0 and x+52=0.

$$x=22 x=-52$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+30x-110=1034$$

Subtract 1034 from both sides.

$$x^{2}+30x-110-1034=0$$

Subtract 1034 from -110 to get -1144.

$$x^{2}+30x-1144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30 for b, and -1144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-1144\right)}}{2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-1144\right)}}{2}$$

Multiply -4 times -1144.

$$x=\frac{-30±\sqrt{900+4576}}{2}$$

Add 900 to 4576.

$$x=\frac{-30±\sqrt{5476}}{2}$$

Take the square root of 5476.

$$x=\frac{-30±74}{2}$$

Now solve the equation x=\frac{-30±74}{2} when ± is plus. Add -30 to 74.

$$x=\frac{44}{2}$$

Divide 44 by 2.

$$x=22$$

Now solve the equation x=\frac{-30±74}{2} when ± is minus. Subtract 74 from -30.

$$x=-\frac{104}{2}$$

Divide -104 by 2.

$$x=-52$$

The equation is now solved.

$$x=22 x=-52$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+30x-110=1034$$

Add 110 to both sides.

$$x^{2}+30x=1034+110$$

Add 1034 and 110 to get 1144.

$$x^{2}+30x=1144$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+30x+15^{2}=1144+15^{2}$$

Square 15.

$$x^{2}+30x+225=1144+225$$

Add 1144 to 225.

$$x^{2}+30x+225=1369$$

Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15\right)^{2}=1369$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{1369}$$

Simplify.

$$x+15=37 x+15=-37$$

Subtract 15 from both sides of the equation.

$$x=22 x=-52$$