Solve 10*10+x+0.01x^2=700

$$10 \cdot 10+x+0.01 { x }^{ 2 } = 700$$

Solve for x

$$x=200$$

$$x=-300$$

Steps Using the Quadratic Formula

Multiply 10 and 10 to get 100.

$$100+x+0.01x^{2}=700$$

Subtract 700 from both sides.

$$100+x+0.01x^{2}-700=0$$

Subtract 700 from 100 to get -600.

$$-600+x+0.01x^{2}=0$$

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.

$$0.01x^{2}+x-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.01 for a, 1 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 0.01\left(-600\right)}}{2\times 0.01}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 0.01\left(-600\right)}}{2\times 0.01}$$

Multiply -4 times 0.01.

$$x=\frac{-1±\sqrt{1-0.04\left(-600\right)}}{2\times 0.01}$$

Multiply -0.04 times -600.

$$x=\frac{-1±\sqrt{1+24}}{2\times 0.01}$$

Add 1 to 24.

$$x=\frac{-1±\sqrt{25}}{2\times 0.01}$$

Take the square root of 25.

$$x=\frac{-1±5}{2\times 0.01}$$

Multiply 2 times 0.01.

$$x=\frac{-1±5}{0.02}$$

Now solve the equation x=\frac{-1±5}{0.02} when ± is plus. Add -1 to 5.

$$x=\frac{4}{0.02}$$

Divide 4 by 0.02 by multiplying 4 by the reciprocal of 0.02.

$$x=200$$

Now solve the equation x=\frac{-1±5}{0.02} when ± is minus. Subtract 5 from -1.

$$x=-\frac{6}{0.02}$$

Divide -6 by 0.02 by multiplying -6 by the reciprocal of 0.02.

$$x=-300$$

The equation is now solved.

$$x=200 x=-300$$

Steps for Completing the Square

Multiply 10 and 10 to get 100.

$$100+x+0.01x^{2}=700$$

Subtract 100 from both sides.

$$x+0.01x^{2}=700-100$$

Subtract 100 from 700 to get 600.

$$x+0.01x^{2}=600$$

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.

$$0.01x^{2}+x=600$$

Multiply both sides by 100.

$$\frac{0.01x^{2}+x}{0.01}=\frac{600}{0.01}$$

Dividing by 0.01 undoes the multiplication by 0.01.

$$x^{2}+\frac{1}{0.01}x=\frac{600}{0.01}$$

Divide 1 by 0.01 by multiplying 1 by the reciprocal of 0.01.

$$x^{2}+100x=\frac{600}{0.01}$$

Divide 600 by 0.01 by multiplying 600 by the reciprocal of 0.01.

$$x^{2}+100x=60000$$

Divide 100, the coefficient of the x term, by 2 to get 50. Then add the square of 50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+100x+50^{2}=60000+50^{2}$$

Square 50.

$$x^{2}+100x+2500=60000+2500$$

Add 60000 to 2500.

$$x^{2}+100x+2500=62500$$

Factor x^{2}+100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+50\right)^{2}=62500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{62500}$$

Simplify.

$$x+50=250 x+50=-250$$

Subtract 50 from both sides of the equation.

$$x=200 x=-300$$