Solve (2500+250x)^2+500^2=(340x)^2

$$( 2500 + 250 x ) ^ { 2 } + 500 ^ { 2 } = ( 340 x ) ^ { 2 }$$

Solve for x

$$x = \frac{50 \sqrt{29431} + 6250}{531} \approx 27.924167233$$

$$x=\frac{6250-50\sqrt{29431}}{531}\approx -4.383677591$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2500+250x\right)^{2}.

$$6250000+1250000x+62500x^{2}+500^{2}=\left(340x\right)^{2}$$

Calculate 500 to the power of 2 and get 250000.

$$6250000+1250000x+62500x^{2}+250000=\left(340x\right)^{2}$$

Add 6250000 and 250000 to get 6500000.

$$6500000+1250000x+62500x^{2}=\left(340x\right)^{2}$$

Expand \left(340x\right)^{2}.

$$6500000+1250000x+62500x^{2}=340^{2}x^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$6500000+1250000x+62500x^{2}=115600x^{2}$$

Subtract 115600x^{2} from both sides.

$$6500000+1250000x+62500x^{2}-115600x^{2}=0$$

Combine 62500x^{2} and -115600x^{2} to get -53100x^{2}.

$$6500000+1250000x-53100x^{2}=0$$

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.

$$-53100x^{2}+1250000x+6500000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -53100 for a, 1250000 for b, and 6500000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1250000±\sqrt{1250000^{2}-4\left(-53100\right)\times 6500000}}{2\left(-53100\right)}$$

Square 1250000.

$$x=\frac{-1250000±\sqrt{1562500000000-4\left(-53100\right)\times 6500000}}{2\left(-53100\right)}$$

Multiply -4 times -53100.

$$x=\frac{-1250000±\sqrt{1562500000000+212400\times 6500000}}{2\left(-53100\right)}$$

Multiply 212400 times 6500000.

$$x=\frac{-1250000±\sqrt{1562500000000+1380600000000}}{2\left(-53100\right)}$$

Add 1562500000000 to 1380600000000.

$$x=\frac{-1250000±\sqrt{2943100000000}}{2\left(-53100\right)}$$

Take the square root of 2943100000000.

$$x=\frac{-1250000±10000\sqrt{29431}}{2\left(-53100\right)}$$

Multiply 2 times -53100.

$$x=\frac{-1250000±10000\sqrt{29431}}{-106200}$$

Now solve the equation x=\frac{-1250000±10000\sqrt{29431}}{-106200} when ± is plus. Add -1250000 to 10000\sqrt{29431}.

$$x=\frac{10000\sqrt{29431}-1250000}{-106200}$$

Divide -1250000+10000\sqrt{29431} by -106200.

$$x=\frac{6250-50\sqrt{29431}}{531}$$

Now solve the equation x=\frac{-1250000±10000\sqrt{29431}}{-106200} when ± is minus. Subtract 10000\sqrt{29431} from -1250000.

$$x=\frac{-10000\sqrt{29431}-1250000}{-106200}$$

Divide -1250000-10000\sqrt{29431} by -106200.

$$x=\frac{50\sqrt{29431}+6250}{531}$$

The equation is now solved.

$$x=\frac{6250-50\sqrt{29431}}{531} x=\frac{50\sqrt{29431}+6250}{531}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2500+250x\right)^{2}.

$$6250000+1250000x+62500x^{2}+500^{2}=\left(340x\right)^{2}$$

Calculate 500 to the power of 2 and get 250000.

$$6250000+1250000x+62500x^{2}+250000=\left(340x\right)^{2}$$

Add 6250000 and 250000 to get 6500000.

$$6500000+1250000x+62500x^{2}=\left(340x\right)^{2}$$

Expand \left(340x\right)^{2}.

$$6500000+1250000x+62500x^{2}=340^{2}x^{2}$$

Calculate 340 to the power of 2 and get 115600.

$$6500000+1250000x+62500x^{2}=115600x^{2}$$

Subtract 115600x^{2} from both sides.

$$6500000+1250000x+62500x^{2}-115600x^{2}=0$$

Combine 62500x^{2} and -115600x^{2} to get -53100x^{2}.

$$6500000+1250000x-53100x^{2}=0$$

Subtract 6500000 from both sides. Anything subtracted from zero gives its negation.

$$1250000x-53100x^{2}=-6500000$$

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.

$$-53100x^{2}+1250000x=-6500000$$

Divide both sides by -53100.

$$\frac{-53100x^{2}+1250000x}{-53100}=-\frac{6500000}{-53100}$$

Dividing by -53100 undoes the multiplication by -53100.

$$x^{2}+\frac{1250000}{-53100}x=-\frac{6500000}{-53100}$$

Reduce the fraction \frac{1250000}{-53100} to lowest terms by extracting and canceling out 100.

$$x^{2}-\frac{12500}{531}x=-\frac{6500000}{-53100}$$

Reduce the fraction \frac{-6500000}{-53100} to lowest terms by extracting and canceling out 100.

$$x^{2}-\frac{12500}{531}x=\frac{65000}{531}$$

Divide -\frac{12500}{531}, the coefficient of the x term, by 2 to get -\frac{6250}{531}. Then add the square of -\frac{6250}{531} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{12500}{531}x+\left(-\frac{6250}{531}\right)^{2}=\frac{65000}{531}+\left(-\frac{6250}{531}\right)^{2}$$

Square -\frac{6250}{531} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{12500}{531}x+\frac{39062500}{281961}=\frac{65000}{531}+\frac{39062500}{281961}$$

Add \frac{65000}{531} to \frac{39062500}{281961} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{12500}{531}x+\frac{39062500}{281961}=\frac{73577500}{281961}$$

Factor x^{2}-\frac{12500}{531}x+\frac{39062500}{281961}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{6250}{531}\right)^{2}=\frac{73577500}{281961}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6250}{531}\right)^{2}}=\sqrt{\frac{73577500}{281961}}$$

Simplify.

$$x-\frac{6250}{531}=\frac{50\sqrt{29431}}{531} x-\frac{6250}{531}=-\frac{50\sqrt{29431}}{531}$$

Add \frac{6250}{531} to both sides of the equation.

$$x=\frac{50\sqrt{29431}+6250}{531} x=\frac{6250-50\sqrt{29431}}{531}$$