Solve -2x^2+14x+600=87

$$- 2 x ^ { 2 } + 14 x + 600 = 87$$

Solve for x

$$x = \frac{5 \sqrt{43} + 7}{2} \approx 19.893596311$$

$$x=\frac{7-5\sqrt{43}}{2}\approx -12.893596311$$

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.

$$-2x^{2}+14x+600=87$$

Subtract 87 from both sides of the equation.

$$-2x^{2}+14x+600-87=87-87$$

Subtracting 87 from itself leaves 0.

$$-2x^{2}+14x+600-87=0$$

Subtract 87 from 600.

$$-2x^{2}+14x+513=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 14 for b, and 513 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-2\right)\times 513}}{2\left(-2\right)}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-2\right)\times 513}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-14±\sqrt{196+8\times 513}}{2\left(-2\right)}$$

Multiply 8 times 513.

$$x=\frac{-14±\sqrt{196+4104}}{2\left(-2\right)}$$

Add 196 to 4104.

$$x=\frac{-14±\sqrt{4300}}{2\left(-2\right)}$$

Take the square root of 4300.

$$x=\frac{-14±10\sqrt{43}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-14±10\sqrt{43}}{-4}$$

Now solve the equation x=\frac{-14±10\sqrt{43}}{-4} when ± is plus. Add -14 to 10\sqrt{43}.

$$x=\frac{10\sqrt{43}-14}{-4}$$

Divide -14+10\sqrt{43} by -4.

$$x=\frac{7-5\sqrt{43}}{2}$$

Now solve the equation x=\frac{-14±10\sqrt{43}}{-4} when ± is minus. Subtract 10\sqrt{43} from -14.

$$x=\frac{-10\sqrt{43}-14}{-4}$$

Divide -14-10\sqrt{43} by -4.

$$x=\frac{5\sqrt{43}+7}{2}$$

The equation is now solved.

$$x=\frac{7-5\sqrt{43}}{2} x=\frac{5\sqrt{43}+7}{2}$$

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.

$$-2x^{2}+14x+600=87$$

Subtract 600 from both sides of the equation.

$$-2x^{2}+14x+600-600=87-600$$

Subtracting 600 from itself leaves 0.

$$-2x^{2}+14x=87-600$$

Subtract 600 from 87.

$$-2x^{2}+14x=-513$$

Divide both sides by -2.

$$\frac{-2x^{2}+14x}{-2}=-\frac{513}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{14}{-2}x=-\frac{513}{-2}$$

Divide 14 by -2.

$$x^{2}-7x=-\frac{513}{-2}$$

Divide -513 by -2.

$$x^{2}-7x=\frac{513}{2}$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=\frac{513}{2}+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=\frac{513}{2}+\frac{49}{4}$$

Add \frac{513}{2} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-7x+\frac{49}{4}=\frac{1075}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{1075}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{1075}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{5\sqrt{43}}{2} x-\frac{7}{2}=-\frac{5\sqrt{43}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{5\sqrt{43}+7}{2} x=\frac{7-5\sqrt{43}}{2}$$