Solve (x+70)^2+x^2=170^2

$$( x + 70 ) ^ { 2 } + x ^ { 2 } = 170 ^ { 2 }$$

Solve for x

$$x=-150$$

$$x=80$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+70\right)^{2}.

$$x^{2}+140x+4900+x^{2}=170^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+140x+4900=170^{2}$$

Calculate 170 to the power of 2 and get 28900.

$$2x^{2}+140x+4900=28900$$

Subtract 28900 from both sides.

$$2x^{2}+140x+4900-28900=0$$

Subtract 28900 from 4900 to get -24000.

$$2x^{2}+140x-24000=0$$

Divide both sides by 2.

$$x^{2}+70x-12000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-12000. To find a and b, set up a system to be solved.

$$a+b=70 ab=1\left(-12000\right)=-12000$$

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 -12000.

$$-1,12000 -2,6000 -3,4000 -4,3000 -5,2400 -6,2000 -8,1500 -10,1200 -12,1000 -15,800 -16,750 -20,600 -24,500 -25,480 -30,400 -32,375 -40,300 -48,250 -50,240 -60,200 -75,160 -80,150 -96,125 -100,120$$

Calculate the sum for each pair.

$$-1+12000=11999 -2+6000=5998 -3+4000=3997 -4+3000=2996 -5+2400=2395 -6+2000=1994 -8+1500=1492 -10+1200=1190 -12+1000=988 -15+800=785 -16+750=734 -20+600=580 -24+500=476 -25+480=455 -30+400=370 -32+375=343 -40+300=260 -48+250=202 -50+240=190 -60+200=140 -75+160=85 -80+150=70 -96+125=29 -100+120=20$$

The solution is the pair that gives sum 70.

$$a=-80 b=150$$

Rewrite x^{2}+70x-12000 as \left(x^{2}-80x\right)+\left(150x-12000\right).

$$\left(x^{2}-80x\right)+\left(150x-12000\right)$$

Factor out x in the first and 150 in the second group.

$$x\left(x-80\right)+150\left(x-80\right)$$

Factor out common term x-80 by using distributive property.

$$\left(x-80\right)\left(x+150\right)$$

To find equation solutions, solve x-80=0 and x+150=0.

$$x=80 x=-150$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+70\right)^{2}.

$$x^{2}+140x+4900+x^{2}=170^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+140x+4900=170^{2}$$

Calculate 170 to the power of 2 and get 28900.

$$2x^{2}+140x+4900=28900$$

Subtract 28900 from both sides.

$$2x^{2}+140x+4900-28900=0$$

Subtract 28900 from 4900 to get -24000.

$$2x^{2}+140x-24000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 140 for b, and -24000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140±\sqrt{140^{2}-4\times 2\left(-24000\right)}}{2\times 2}$$

Square 140.

$$x=\frac{-140±\sqrt{19600-4\times 2\left(-24000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-140±\sqrt{19600-8\left(-24000\right)}}{2\times 2}$$

Multiply -8 times -24000.

$$x=\frac{-140±\sqrt{19600+192000}}{2\times 2}$$

Add 19600 to 192000.

$$x=\frac{-140±\sqrt{211600}}{2\times 2}$$

Take the square root of 211600.

$$x=\frac{-140±460}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-140±460}{4}$$

Now solve the equation x=\frac{-140±460}{4} when ± is plus. Add -140 to 460.

$$x=\frac{320}{4}$$

Divide 320 by 4.

$$x=80$$

Now solve the equation x=\frac{-140±460}{4} when ± is minus. Subtract 460 from -140.

$$x=-\frac{600}{4}$$

Divide -600 by 4.

$$x=-150$$

The equation is now solved.

$$x=80 x=-150$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+70\right)^{2}.

$$x^{2}+140x+4900+x^{2}=170^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+140x+4900=170^{2}$$

Calculate 170 to the power of 2 and get 28900.

$$2x^{2}+140x+4900=28900$$

Subtract 4900 from both sides.

$$2x^{2}+140x=28900-4900$$

Subtract 4900 from 28900 to get 24000.

$$2x^{2}+140x=24000$$

Divide both sides by 2.

$$\frac{2x^{2}+140x}{2}=\frac{24000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{140}{2}x=\frac{24000}{2}$$

Divide 140 by 2.

$$x^{2}+70x=\frac{24000}{2}$$

Divide 24000 by 2.

$$x^{2}+70x=12000$$

Divide 70, the coefficient of the x term, by 2 to get 35. Then add the square of 35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+70x+35^{2}=12000+35^{2}$$

Square 35.

$$x^{2}+70x+1225=12000+1225$$

Add 12000 to 1225.

$$x^{2}+70x+1225=13225$$

Factor x^{2}+70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+35\right)^{2}=13225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+35\right)^{2}}=\sqrt{13225}$$

Simplify.

$$x+35=115 x+35=-115$$

Subtract 35 from both sides of the equation.

$$x=80 x=-150$$