Solve x^2+17x-17750=0

$${ x }^{ 2 } +17x-17750 = 0$$

Solve for x

$$x=-142$$

$$x=125$$

Steps Using Factoring

To solve the equation, factor x^{2}+17x-17750 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.

$$a+b=17 ab=-17750$$

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 -17750.

$$-1,17750 -2,8875 -5,3550 -10,1775 -25,710 -50,355 -71,250 -125,142$$

Calculate the sum for each pair.

$$-1+17750=17749 -2+8875=8873 -5+3550=3545 -10+1775=1765 -25+710=685 -50+355=305 -71+250=179 -125+142=17$$

The solution is the pair that gives sum 17.

$$a=-125 b=142$$

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

$$\left(x-125\right)\left(x+142\right)$$

To find equation solutions, solve x-125=0 and x+142=0.

$$x=125 x=-142$$

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-17750. To find a and b, set up a system to be solved.

$$a+b=17 ab=1\left(-17750\right)=-17750$$

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 -17750.

$$-1,17750 -2,8875 -5,3550 -10,1775 -25,710 -50,355 -71,250 -125,142$$

Calculate the sum for each pair.

$$-1+17750=17749 -2+8875=8873 -5+3550=3545 -10+1775=1765 -25+710=685 -50+355=305 -71+250=179 -125+142=17$$

The solution is the pair that gives sum 17.

$$a=-125 b=142$$

Rewrite x^{2}+17x-17750 as \left(x^{2}-125x\right)+\left(142x-17750\right).

$$\left(x^{2}-125x\right)+\left(142x-17750\right)$$

Factor out x in the first and 142 in the second group.

$$x\left(x-125\right)+142\left(x-125\right)$$

Factor out common term x-125 by using distributive property.

$$\left(x-125\right)\left(x+142\right)$$

To find equation solutions, solve x-125=0 and x+142=0.

$$x=125 x=-142$$

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.

$$x^{2}+17x-17750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 17 for b, and -17750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-17750\right)}}{2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-17750\right)}}{2}$$

Multiply -4 times -17750.

$$x=\frac{-17±\sqrt{289+71000}}{2}$$

Add 289 to 71000.

$$x=\frac{-17±\sqrt{71289}}{2}$$

Take the square root of 71289.

$$x=\frac{-17±267}{2}$$

Now solve the equation x=\frac{-17±267}{2} when ± is plus. Add -17 to 267.

$$x=\frac{250}{2}$$

Divide 250 by 2.

$$x=125$$

Now solve the equation x=\frac{-17±267}{2} when ± is minus. Subtract 267 from -17.

$$x=-\frac{284}{2}$$

Divide -284 by 2.

$$x=-142$$

The equation is now solved.

$$x=125 x=-142$$

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.

$$x^{2}+17x-17750=0$$

Add 17750 to both sides of the equation.

$$x^{2}+17x-17750-\left(-17750\right)=-\left(-17750\right)$$

Subtracting -17750 from itself leaves 0.

$$x^{2}+17x=-\left(-17750\right)$$

Subtract -17750 from 0.

$$x^{2}+17x=17750$$

Divide 17, the coefficient of the x term, by 2 to get \frac{17}{2}. Then add the square of \frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+17x+\left(\frac{17}{2}\right)^{2}=17750+\left(\frac{17}{2}\right)^{2}$$

Square \frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+17x+\frac{289}{4}=17750+\frac{289}{4}$$

Add 17750 to \frac{289}{4}.

$$x^{2}+17x+\frac{289}{4}=\frac{71289}{4}$$

Factor x^{2}+17x+\frac{289}{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}.

$$\left(x+\frac{17}{2}\right)^{2}=\frac{71289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{2}\right)^{2}}=\sqrt{\frac{71289}{4}}$$

Simplify.

$$x+\frac{17}{2}=\frac{267}{2} x+\frac{17}{2}=-\frac{267}{2}$$

Subtract \frac{17}{2} from both sides of the equation.

$$x=125 x=-142$$