Solve -2x^2+58x=200

$$- 2 x ^ { 2 } + 58 x = 200$$

Solve for x

$$x=4$$

$$x=25$$

Steps Using Factoring By Grouping

Subtract 200 from both sides.

$$-2x^{2}+58x-200=0$$

Divide both sides by 2.

$$-x^{2}+29x-100=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-100. To find a and b, set up a system to be solved.

$$a+b=29 ab=-\left(-100\right)=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Calculate the sum for each pair.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

The solution is the pair that gives sum 29.

$$a=25 b=4$$

Rewrite -x^{2}+29x-100 as \left(-x^{2}+25x\right)+\left(4x-100\right).

$$\left(-x^{2}+25x\right)+\left(4x-100\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-25\right)+4\left(x-25\right)$$

Factor out common term x-25 by using distributive property.

$$\left(x-25\right)\left(-x+4\right)$$

To find equation solutions, solve x-25=0 and -x+4=0.

$$x=25 x=4$$

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.

$$-2x^{2}+58x=200$$

Subtract 200 from both sides of the equation.

$$-2x^{2}+58x-200=200-200$$

Subtracting 200 from itself leaves 0.

$$-2x^{2}+58x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 58 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-58±\sqrt{58^{2}-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Square 58.

$$x=\frac{-58±\sqrt{3364-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-58±\sqrt{3364+8\left(-200\right)}}{2\left(-2\right)}$$

Multiply 8 times -200.

$$x=\frac{-58±\sqrt{3364-1600}}{2\left(-2\right)}$$

Add 3364 to -1600.

$$x=\frac{-58±\sqrt{1764}}{2\left(-2\right)}$$

Take the square root of 1764.

$$x=\frac{-58±42}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-58±42}{-4}$$

Now solve the equation x=\frac{-58±42}{-4} when ± is plus. Add -58 to 42.

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

Now solve the equation x=\frac{-58±42}{-4} when ± is minus. Subtract 42 from -58.

$$x=-\frac{100}{-4}$$

Divide -100 by -4.

$$x=25$$

The equation is now solved.

$$x=4 x=25$$

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.

$$-2x^{2}+58x=200$$

Divide both sides by -2.

$$\frac{-2x^{2}+58x}{-2}=\frac{200}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{58}{-2}x=\frac{200}{-2}$$

Divide 58 by -2.

$$x^{2}-29x=\frac{200}{-2}$$

Divide 200 by -2.

$$x^{2}-29x=-100$$

Divide -29, the coefficient of the x term, by 2 to get -\frac{29}{2}. Then add the square of -\frac{29}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-29x+\left(-\frac{29}{2}\right)^{2}=-100+\left(-\frac{29}{2}\right)^{2}$$

Square -\frac{29}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-29x+\frac{841}{4}=-100+\frac{841}{4}$$

Add -100 to \frac{841}{4}.

$$x^{2}-29x+\frac{841}{4}=\frac{441}{4}$$

Factor x^{2}-29x+\frac{841}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{2}\right)^{2}=\frac{441}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{2}\right)^{2}}=\sqrt{\frac{441}{4}}$$

Simplify.

$$x-\frac{29}{2}=\frac{21}{2} x-\frac{29}{2}=-\frac{21}{2}$$

Add \frac{29}{2} to both sides of the equation.

$$x=25 x=4$$