Solve x^2+34x+240=0

$${ x }^{ 2 } +34x+240=0$$

Solve for x

$$x=-24$$

$$x=-10$$

Steps Using Factoring

To solve the equation, factor x^{2}+34x+240 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.

$$a+b=34 ab=240$$

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 240.

$$1,240 2,120 3,80 4,60 5,48 6,40 8,30 10,24 12,20 15,16$$

Calculate the sum for each pair.

$$1+240=241 2+120=122 3+80=83 4+60=64 5+48=53 6+40=46 8+30=38 10+24=34 12+20=32 15+16=31$$

The solution is the pair that gives sum 34.

$$a=10 b=24$$

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

$$\left(x+10\right)\left(x+24\right)$$

To find equation solutions, solve x+10=0 and x+24=0.

$$x=-10 x=-24$$

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+240. To find a and b, set up a system to be solved.

$$a+b=34 ab=1\times 240=240$$

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 240.

$$1,240 2,120 3,80 4,60 5,48 6,40 8,30 10,24 12,20 15,16$$

Calculate the sum for each pair.

$$1+240=241 2+120=122 3+80=83 4+60=64 5+48=53 6+40=46 8+30=38 10+24=34 12+20=32 15+16=31$$

The solution is the pair that gives sum 34.

$$a=10 b=24$$

Rewrite x^{2}+34x+240 as \left(x^{2}+10x\right)+\left(24x+240\right).

$$\left(x^{2}+10x\right)+\left(24x+240\right)$$

Factor out x in the first and 24 in the second group.

$$x\left(x+10\right)+24\left(x+10\right)$$

Factor out common term x+10 by using distributive property.

$$\left(x+10\right)\left(x+24\right)$$

To find equation solutions, solve x+10=0 and x+24=0.

$$x=-10 x=-24$$

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.

$$x^{2}+34x+240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 34 for b, and 240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\times 240}}{2}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\times 240}}{2}$$

Multiply -4 times 240.

$$x=\frac{-34±\sqrt{1156-960}}{2}$$

Add 1156 to -960.

$$x=\frac{-34±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-34±14}{2}$$

Now solve the equation x=\frac{-34±14}{2} when ± is plus. Add -34 to 14.

$$x=-\frac{20}{2}$$

Divide -20 by 2.

$$x=-10$$

Now solve the equation x=\frac{-34±14}{2} when ± is minus. Subtract 14 from -34.

$$x=-\frac{48}{2}$$

Divide -48 by 2.

$$x=-24$$

The equation is now solved.

$$x=-10 x=-24$$

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.

$$x^{2}+34x+240=0$$

Subtract 240 from both sides of the equation.

$$x^{2}+34x+240-240=-240$$

Subtracting 240 from itself leaves 0.

$$x^{2}+34x=-240$$

Divide 34, the coefficient of the x term, by 2 to get 17. Then add the square of 17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+34x+17^{2}=-240+17^{2}$$

Square 17.

$$x^{2}+34x+289=-240+289$$

Add -240 to 289.

$$x^{2}+34x+289=49$$

Factor x^{2}+34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+17\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{49}$$

Simplify.

$$x+17=7 x+17=-7$$

Subtract 17 from both sides of the equation.

$$x=-10 x=-24$$