Solve x^2-30x-1000=0

$${ x }^{ 2 } -30x-1000=0$$

Solve for x

$$x=-20$$

$$x=50$$

Steps Using Factoring

To solve the equation, factor x^{2}-30x-1000 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.

$$a+b=-30 ab=-1000$$

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 -1000.

$$1,-1000 2,-500 4,-250 5,-200 8,-125 10,-100 20,-50 25,-40$$

Calculate the sum for each pair.

$$1-1000=-999 2-500=-498 4-250=-246 5-200=-195 8-125=-117 10-100=-90 20-50=-30 25-40=-15$$

The solution is the pair that gives sum -30.

$$a=-50 b=20$$

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

$$\left(x-50\right)\left(x+20\right)$$

To find equation solutions, solve x-50=0 and x+20=0.

$$x=50 x=-20$$

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-1000. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\left(-1000\right)=-1000$$

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 -1000.

$$1,-1000 2,-500 4,-250 5,-200 8,-125 10,-100 20,-50 25,-40$$

Calculate the sum for each pair.

$$1-1000=-999 2-500=-498 4-250=-246 5-200=-195 8-125=-117 10-100=-90 20-50=-30 25-40=-15$$

The solution is the pair that gives sum -30.

$$a=-50 b=20$$

Rewrite x^{2}-30x-1000 as \left(x^{2}-50x\right)+\left(20x-1000\right).

$$\left(x^{2}-50x\right)+\left(20x-1000\right)$$

Factor out x in the first and 20 in the second group.

$$x\left(x-50\right)+20\left(x-50\right)$$

Factor out common term x-50 by using distributive property.

$$\left(x-50\right)\left(x+20\right)$$

To find equation solutions, solve x-50=0 and x+20=0.

$$x=50 x=-20$$

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.

$$x^{2}-30x-1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -30 for b, and -1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-1000\right)}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\left(-1000\right)}}{2}$$

Multiply -4 times -1000.

$$x=\frac{-\left(-30\right)±\sqrt{900+4000}}{2}$$

Add 900 to 4000.

$$x=\frac{-\left(-30\right)±\sqrt{4900}}{2}$$

Take the square root of 4900.

$$x=\frac{-\left(-30\right)±70}{2}$$

The opposite of -30 is 30.

$$x=\frac{30±70}{2}$$

Now solve the equation x=\frac{30±70}{2} when ± is plus. Add 30 to 70.

$$x=\frac{100}{2}$$

Divide 100 by 2.

$$x=50$$

Now solve the equation x=\frac{30±70}{2} when ± is minus. Subtract 70 from 30.

$$x=-\frac{40}{2}$$

Divide -40 by 2.

$$x=-20$$

The equation is now solved.

$$x=50 x=-20$$

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.

$$x^{2}-30x-1000=0$$

Add 1000 to both sides of the equation.

$$x^{2}-30x-1000-\left(-1000\right)=-\left(-1000\right)$$

Subtracting -1000 from itself leaves 0.

$$x^{2}-30x=-\left(-1000\right)$$

Subtract -1000 from 0.

$$x^{2}-30x=1000$$

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.

$$x^{2}-30x+\left(-15\right)^{2}=1000+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=1000+225$$

Add 1000 to 225.

$$x^{2}-30x+225=1225$$

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}.

$$\left(x-15\right)^{2}=1225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{1225}$$

Simplify.

$$x-15=35 x-15=-35$$

Add 15 to both sides of the equation.

$$x=50 x=-20$$