Solve -30w-25=9w^2

$$- 30 w - 25 = 9 w ^ { 2 }$$

Solve for w

$$w = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

Steps Using Factoring By Grouping

Subtract 9w^{2} from both sides.

$$-30w-25-9w^{2}=0$$

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

$$-9w^{2}-30w-25=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9w^{2}+aw+bw-25. To find a and b, set up a system to be solved.

$$a+b=-30 ab=-9\left(-25\right)=225$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 -30.

$$a=-15 b=-15$$

Rewrite -9w^{2}-30w-25 as \left(-9w^{2}-15w\right)+\left(-15w-25\right).

$$\left(-9w^{2}-15w\right)+\left(-15w-25\right)$$

Factor out 3w in the first and 5 in the second group.

$$3w\left(-3w-5\right)+5\left(-3w-5\right)$$

Factor out common term -3w-5 by using distributive property.

$$\left(-3w-5\right)\left(3w+5\right)$$

To find equation solutions, solve -3w-5=0 and 3w+5=0.

$$w=-\frac{5}{3} w=-\frac{5}{3}$$

Steps Using the Quadratic Formula

Subtract 9w^{2} from both sides.

$$-30w-25-9w^{2}=0$$

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.

$$-9w^{2}-30w-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -30 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\left(-9\right)\left(-25\right)}}{2\left(-9\right)}$$

Square -30.

$$w=\frac{-\left(-30\right)±\sqrt{900-4\left(-9\right)\left(-25\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$w=\frac{-\left(-30\right)±\sqrt{900+36\left(-25\right)}}{2\left(-9\right)}$$

Multiply 36 times -25.

$$w=\frac{-\left(-30\right)±\sqrt{900-900}}{2\left(-9\right)}$$

Add 900 to -900.

$$w=\frac{-\left(-30\right)±\sqrt{0}}{2\left(-9\right)}$$

Take the square root of 0.

$$w=-\frac{-30}{2\left(-9\right)}$$

The opposite of -30 is 30.

$$w=\frac{30}{2\left(-9\right)}$$

Multiply 2 times -9.

$$w=\frac{30}{-18}$$

Reduce the fraction \frac{30}{-18} to lowest terms by extracting and canceling out 6.

$$w=-\frac{5}{3}$$

Steps for Completing the Square

Subtract 9w^{2} from both sides.

$$-30w-25-9w^{2}=0$$

Add 25 to both sides. Anything plus zero gives itself.

$$-30w-9w^{2}=25$$

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.

$$-9w^{2}-30w=25$$

Divide both sides by -9.

$$\frac{-9w^{2}-30w}{-9}=\frac{25}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$w^{2}+\left(-\frac{30}{-9}\right)w=\frac{25}{-9}$$

Reduce the fraction \frac{-30}{-9} to lowest terms by extracting and canceling out 3.

$$w^{2}+\frac{10}{3}w=\frac{25}{-9}$$

Divide 25 by -9.

$$w^{2}+\frac{10}{3}w=-\frac{25}{9}$$

Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+\frac{10}{3}w+\left(\frac{5}{3}\right)^{2}=-\frac{25}{9}+\left(\frac{5}{3}\right)^{2}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{-25+25}{9}$$

Add -\frac{25}{9} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=0$$

Factor w^{2}+\frac{10}{3}w+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+\frac{5}{3}\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{5}{3}\right)^{2}}=\sqrt{0}$$

Simplify.

$$w+\frac{5}{3}=0 w+\frac{5}{3}=0$$

Subtract \frac{5}{3} from both sides of the equation.

$$w=-\frac{5}{3} w=-\frac{5}{3}$$

The equation is now solved. Solutions are the same.

$$w=-\frac{5}{3}$$