Solve -2x^2+100x=450

$$-2 { x }^{ 2 } +100x=450$$

Solve for x

$$x=5$$

$$x=45$$

Steps Using Factoring By Grouping

Subtract 450 from both sides.

$$-2x^{2}+100x-450=0$$

Divide both sides by 2.

$$-x^{2}+50x-225=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-225. To find a and b, set up a system to be solved.

$$a+b=50 ab=-\left(-225\right)=225$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 225.

$$1,225 3,75 5,45 9,25 15,15$$

Calculate the sum for each pair.

$$1+225=226 3+75=78 5+45=50 9+25=34 15+15=30$$

The solution is the pair that gives sum 50.

$$a=45 b=5$$

Rewrite -x^{2}+50x-225 as \left(-x^{2}+45x\right)+\left(5x-225\right).

$$\left(-x^{2}+45x\right)+\left(5x-225\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-45\right)+5\left(x-45\right)$$

Factor out common term x-45 by using distributive property.

$$\left(x-45\right)\left(-x+5\right)$$

To find equation solutions, solve x-45=0 and -x+5=0.

$$x=45 x=5$$

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.

$$-2x^{2}+100x=450$$

Subtract 450 from both sides of the equation.

$$-2x^{2}+100x-450=450-450$$

Subtracting 450 from itself leaves 0.

$$-2x^{2}+100x-450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 100 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-100±\sqrt{10000+8\left(-450\right)}}{2\left(-2\right)}$$

Multiply 8 times -450.

$$x=\frac{-100±\sqrt{10000-3600}}{2\left(-2\right)}$$

Add 10000 to -3600.

$$x=\frac{-100±\sqrt{6400}}{2\left(-2\right)}$$

Take the square root of 6400.

$$x=\frac{-100±80}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-100±80}{-4}$$

Now solve the equation x=\frac{-100±80}{-4} when ± is plus. Add -100 to 80.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-100±80}{-4} when ± is minus. Subtract 80 from -100.

$$x=-\frac{180}{-4}$$

Divide -180 by -4.

$$x=45$$

The equation is now solved.

$$x=5 x=45$$

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.

$$-2x^{2}+100x=450$$

Divide both sides by -2.

$$\frac{-2x^{2}+100x}{-2}=\frac{450}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{100}{-2}x=\frac{450}{-2}$$

Divide 100 by -2.

$$x^{2}-50x=\frac{450}{-2}$$

Divide 450 by -2.

$$x^{2}-50x=-225$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-225+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-225+625$$

Add -225 to 625.

$$x^{2}-50x+625=400$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-25=20 x-25=-20$$

Add 25 to both sides of the equation.

$$x=45 x=5$$