Solve -9*x^2-63*x-54=0

$$- 9 \cdot x ^ { 2 } - 63 \cdot x - 54 = 0$$

Solve for x

$$x=-6$$

$$x=-1$$

Steps Using Factoring By Grouping

Divide both sides by 9.

$$-x^{2}-7x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-6\right)=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-1 b=-6$$

Rewrite -x^{2}-7x-6 as \left(-x^{2}-x\right)+\left(-6x-6\right).

$$\left(-x^{2}-x\right)+\left(-6x-6\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(-x-1\right)+6\left(-x-1\right)$$

Factor out common term -x-1 by using distributive property.

$$\left(-x-1\right)\left(x+6\right)$$

To find equation solutions, solve -x-1=0 and x+6=0.

$$x=-1 x=-6$$

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.

$$-9x^{2}-63x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -63 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}-4\left(-9\right)\left(-54\right)}}{2\left(-9\right)}$$

Square -63.

$$x=\frac{-\left(-63\right)±\sqrt{3969-4\left(-9\right)\left(-54\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-63\right)±\sqrt{3969+36\left(-54\right)}}{2\left(-9\right)}$$

Multiply 36 times -54.

$$x=\frac{-\left(-63\right)±\sqrt{3969-1944}}{2\left(-9\right)}$$

Add 3969 to -1944.

$$x=\frac{-\left(-63\right)±\sqrt{2025}}{2\left(-9\right)}$$

Take the square root of 2025.

$$x=\frac{-\left(-63\right)±45}{2\left(-9\right)}$$

The opposite of -63 is 63.

$$x=\frac{63±45}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{63±45}{-18}$$

Now solve the equation x=\frac{63±45}{-18} when ± is plus. Add 63 to 45.

$$x=\frac{108}{-18}$$

Divide 108 by -18.

$$x=-6$$

Now solve the equation x=\frac{63±45}{-18} when ± is minus. Subtract 45 from 63.

$$x=\frac{18}{-18}$$

Divide 18 by -18.

$$x=-1$$

The equation is now solved.

$$x=-6 x=-1$$

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.

$$-9x^{2}-63x-54=0$$

Add 54 to both sides of the equation.

$$-9x^{2}-63x-54-\left(-54\right)=-\left(-54\right)$$

Subtracting -54 from itself leaves 0.

$$-9x^{2}-63x=-\left(-54\right)$$

Subtract -54 from 0.

$$-9x^{2}-63x=54$$

Divide both sides by -9.

$$\frac{-9x^{2}-63x}{-9}=\frac{54}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{63}{-9}\right)x=\frac{54}{-9}$$

Divide -63 by -9.

$$x^{2}+7x=\frac{54}{-9}$$

Divide 54 by -9.

$$x^{2}+7x=-6$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-6+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=-6+\frac{49}{4}$$

Add -6 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{25}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

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

Subtract \frac{7}{2} from both sides of the equation.

$$x=-1 x=-6$$