Solve -10x^2+1450x+10000=29500

$$-10 { x }^{ 2 } +1450x+10000 = 29500$$

Solve for x

$$x=15$$

$$x=130$$

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.

$$-10x^{2}+1450x+10000=29500$$

Subtract 29500 from both sides of the equation.

$$-10x^{2}+1450x+10000-29500=29500-29500$$

Subtracting 29500 from itself leaves 0.

$$-10x^{2}+1450x+10000-29500=0$$

Subtract 29500 from 10000.

$$-10x^{2}+1450x-19500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1450 for b, and -19500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1450±\sqrt{1450^{2}-4\left(-10\right)\left(-19500\right)}}{2\left(-10\right)}$$

Square 1450.

$$x=\frac{-1450±\sqrt{2102500-4\left(-10\right)\left(-19500\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1450±\sqrt{2102500+40\left(-19500\right)}}{2\left(-10\right)}$$

Multiply 40 times -19500.

$$x=\frac{-1450±\sqrt{2102500-780000}}{2\left(-10\right)}$$

Add 2102500 to -780000.

$$x=\frac{-1450±\sqrt{1322500}}{2\left(-10\right)}$$

Take the square root of 1322500.

$$x=\frac{-1450±1150}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1450±1150}{-20}$$

Now solve the equation x=\frac{-1450±1150}{-20} when ± is plus. Add -1450 to 1150.

$$x=-\frac{300}{-20}$$

Divide -300 by -20.

$$x=15$$

Now solve the equation x=\frac{-1450±1150}{-20} when ± is minus. Subtract 1150 from -1450.

$$x=-\frac{2600}{-20}$$

Divide -2600 by -20.

$$x=130$$

The equation is now solved.

$$x=15 x=130$$

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.

$$-10x^{2}+1450x+10000=29500$$

Subtract 10000 from both sides of the equation.

$$-10x^{2}+1450x+10000-10000=29500-10000$$

Subtracting 10000 from itself leaves 0.

$$-10x^{2}+1450x=29500-10000$$

Subtract 10000 from 29500.

$$-10x^{2}+1450x=19500$$

Divide both sides by -10.

$$\frac{-10x^{2}+1450x}{-10}=\frac{19500}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1450}{-10}x=\frac{19500}{-10}$$

Divide 1450 by -10.

$$x^{2}-145x=\frac{19500}{-10}$$

Divide 19500 by -10.

$$x^{2}-145x=-1950$$

Divide -145, the coefficient of the x term, by 2 to get -\frac{145}{2}. Then add the square of -\frac{145}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-145x+\left(-\frac{145}{2}\right)^{2}=-1950+\left(-\frac{145}{2}\right)^{2}$$

Square -\frac{145}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-145x+\frac{21025}{4}=-1950+\frac{21025}{4}$$

Add -1950 to \frac{21025}{4}.

$$x^{2}-145x+\frac{21025}{4}=\frac{13225}{4}$$

Factor x^{2}-145x+\frac{21025}{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}.

$$\left(x-\frac{145}{2}\right)^{2}=\frac{13225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{145}{2}\right)^{2}}=\sqrt{\frac{13225}{4}}$$

Simplify.

$$x-\frac{145}{2}=\frac{115}{2} x-\frac{145}{2}=-\frac{115}{2}$$

Add \frac{145}{2} to both sides of the equation.

$$x=130 x=15$$