Solve x*(-20x+920)=3100

$$x \cdot ( - 20 x + 920 ) = 3100$$

Solve for x

$$x=\sqrt{374}+23\approx 42.339079606$$

$$x=23-\sqrt{374}\approx 3.660920394$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by -20x+920.

$$-20x^{2}+920x=3100$$

Subtract 3100 from both sides.

$$-20x^{2}+920x-3100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 920 for b, and -3100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-920±\sqrt{920^{2}-4\left(-20\right)\left(-3100\right)}}{2\left(-20\right)}$$

Square 920.

$$x=\frac{-920±\sqrt{846400-4\left(-20\right)\left(-3100\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-920±\sqrt{846400+80\left(-3100\right)}}{2\left(-20\right)}$$

Multiply 80 times -3100.

$$x=\frac{-920±\sqrt{846400-248000}}{2\left(-20\right)}$$

Add 846400 to -248000.

$$x=\frac{-920±\sqrt{598400}}{2\left(-20\right)}$$

Take the square root of 598400.

$$x=\frac{-920±40\sqrt{374}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-920±40\sqrt{374}}{-40}$$

Now solve the equation x=\frac{-920±40\sqrt{374}}{-40} when ± is plus. Add -920 to 40\sqrt{374}.

$$x=\frac{40\sqrt{374}-920}{-40}$$

Divide -920+40\sqrt{374} by -40.

$$x=23-\sqrt{374}$$

Now solve the equation x=\frac{-920±40\sqrt{374}}{-40} when ± is minus. Subtract 40\sqrt{374} from -920.

$$x=\frac{-40\sqrt{374}-920}{-40}$$

Divide -920-40\sqrt{374} by -40.

$$x=\sqrt{374}+23$$

The equation is now solved.

$$x=23-\sqrt{374} x=\sqrt{374}+23$$

Steps for Completing the Square

Use the distributive property to multiply x by -20x+920.

$$-20x^{2}+920x=3100$$

Divide both sides by -20.

$$\frac{-20x^{2}+920x}{-20}=\frac{3100}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{920}{-20}x=\frac{3100}{-20}$$

Divide 920 by -20.

$$x^{2}-46x=\frac{3100}{-20}$$

Divide 3100 by -20.

$$x^{2}-46x=-155$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=-155+\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=-155+529$$

Add -155 to 529.

$$x^{2}-46x+529=374$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=374$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{374}$$

Simplify.

$$x-23=\sqrt{374} x-23=-\sqrt{374}$$

Add 23 to both sides of the equation.

$$x=\sqrt{374}+23 x=23-\sqrt{374}$$