Solve x^2/-17-60/17=-x

$$\frac { x ^ { 2 } } { - 17 } - \frac { 60 } { 17 } = - x$$

Solve for x

$$x=5$$

$$x=12$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 17, the least common multiple of -17,17.

$$-x^{2}-60=-17x$$

Add 17x to both sides.

$$-x^{2}-60+17x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+17x-60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-60. To find a and b, set up a system to be solved.

$$a+b=17 ab=-\left(-60\right)=60$$

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 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 17.

$$a=12 b=5$$

Rewrite -x^{2}+17x-60 as \left(-x^{2}+12x\right)+\left(5x-60\right).

$$\left(-x^{2}+12x\right)+\left(5x-60\right)$$

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

$$-x\left(x-12\right)+5\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x+5\right)$$

To find equation solutions, solve x-12=0 and -x+5=0.

$$x=12 x=5$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 17, the least common multiple of -17,17.

$$-x^{2}-60=-17x$$

Add 17x to both sides.

$$-x^{2}-60+17x=0$$

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.

$$-x^{2}+17x-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 17 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-17±\sqrt{289+4\left(-60\right)}}{2\left(-1\right)}$$

Multiply 4 times -60.

$$x=\frac{-17±\sqrt{289-240}}{2\left(-1\right)}$$

Add 289 to -240.

$$x=\frac{-17±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-17±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-17±7}{-2}$$

Now solve the equation x=\frac{-17±7}{-2} when ± is plus. Add -17 to 7.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-17±7}{-2} when ± is minus. Subtract 7 from -17.

$$x=-\frac{24}{-2}$$

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=5 x=12$$

Steps for Completing the Square

Multiply both sides of the equation by 17, the least common multiple of -17,17.

$$-x^{2}-60=-17x$$

Add 17x to both sides.

$$-x^{2}-60+17x=0$$

Add 60 to both sides. Anything plus zero gives itself.

$$-x^{2}+17x=60$$

Divide both sides by -1.

$$\frac{-x^{2}+17x}{-1}=\frac{60}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{17}{-1}x=\frac{60}{-1}$$

Divide 17 by -1.

$$x^{2}-17x=\frac{60}{-1}$$

Divide 60 by -1.

$$x^{2}-17x=-60$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-60+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-60+\frac{289}{4}$$

Add -60 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{49}{4}$$

Factor x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{7}{2} x-\frac{17}{2}=-\frac{7}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=12 x=5$$