Solve 0.7x^2-525x+1500=0

$$0.7 { x }^{ 2 } -525x+1500=0$$

Solve for x

$$x=\frac{25\sqrt{10857}}{7}+375\approx 747.131889062$$

$$x=-\frac{25\sqrt{10857}}{7}+375\approx 2.868110938$$

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.

$$0.7x^{2}-525x+1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.7 for a, -525 for b, and 1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-525\right)±\sqrt{\left(-525\right)^{2}-4\times 0.7\times 1500}}{2\times 0.7}$$

Square -525.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4\times 0.7\times 1500}}{2\times 0.7}$$

Multiply -4 times 0.7.

$$x=\frac{-\left(-525\right)±\sqrt{275625-2.8\times 1500}}{2\times 0.7}$$

Multiply -2.8 times 1500.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4200}}{2\times 0.7}$$

Add 275625 to -4200.

$$x=\frac{-\left(-525\right)±\sqrt{271425}}{2\times 0.7}$$

Take the square root of 271425.

$$x=\frac{-\left(-525\right)±5\sqrt{10857}}{2\times 0.7}$$

The opposite of -525 is 525.

$$x=\frac{525±5\sqrt{10857}}{2\times 0.7}$$

Multiply 2 times 0.7.

$$x=\frac{525±5\sqrt{10857}}{1.4}$$

Now solve the equation x=\frac{525±5\sqrt{10857}}{1.4} when ± is plus. Add 525 to 5\sqrt{10857}.

$$x=\frac{5\sqrt{10857}+525}{1.4}$$

Divide 525+5\sqrt{10857} by 1.4 by multiplying 525+5\sqrt{10857} by the reciprocal of 1.4.

$$x=\frac{25\sqrt{10857}}{7}+375$$

Now solve the equation x=\frac{525±5\sqrt{10857}}{1.4} when ± is minus. Subtract 5\sqrt{10857} from 525.

$$x=\frac{525-5\sqrt{10857}}{1.4}$$

Divide 525-5\sqrt{10857} by 1.4 by multiplying 525-5\sqrt{10857} by the reciprocal of 1.4.

$$x=-\frac{25\sqrt{10857}}{7}+375$$

The equation is now solved.

$$x=\frac{25\sqrt{10857}}{7}+375 x=-\frac{25\sqrt{10857}}{7}+375$$

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.

$$0.7x^{2}-525x+1500=0$$

Subtract 1500 from both sides of the equation.

$$0.7x^{2}-525x+1500-1500=-1500$$

Subtracting 1500 from itself leaves 0.

$$0.7x^{2}-525x=-1500$$

Divide both sides of the equation by 0.7, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.7x^{2}-525x}{0.7}=-\frac{1500}{0.7}$$

Dividing by 0.7 undoes the multiplication by 0.7.

$$x^{2}+\left(-\frac{525}{0.7}\right)x=-\frac{1500}{0.7}$$

Divide -525 by 0.7 by multiplying -525 by the reciprocal of 0.7.

$$x^{2}-750x=-\frac{1500}{0.7}$$

Divide -1500 by 0.7 by multiplying -1500 by the reciprocal of 0.7.

$$x^{2}-750x=-\frac{15000}{7}$$

Divide -750, the coefficient of the x term, by 2 to get -375. Then add the square of -375 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-750x+\left(-375\right)^{2}=-\frac{15000}{7}+\left(-375\right)^{2}$$

Square -375.

$$x^{2}-750x+140625=-\frac{15000}{7}+140625$$

Add -\frac{15000}{7} to 140625.

$$x^{2}-750x+140625=\frac{969375}{7}$$

Factor x^{2}-750x+140625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-375\right)^{2}=\frac{969375}{7}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-375\right)^{2}}=\sqrt{\frac{969375}{7}}$$

Simplify.

$$x-375=\frac{25\sqrt{10857}}{7} x-375=-\frac{25\sqrt{10857}}{7}$$

Add 375 to both sides of the equation.

$$x=\frac{25\sqrt{10857}}{7}+375 x=-\frac{25\sqrt{10857}}{7}+375$$